In this talk, we give an overview of several recent results where quantitative estimates play a key role. In the first part of the talk, we discuss convex integration constructions for fluid systems with external forcing. We will then discuss a novel application of these ideas to the surface quasi-geostrophic (SQG) equation. Moving forward with the theme of quantitative estimates, in the second part of the talk we will describe new bounds for the defocusing energy-supercritical Nonlinear Schrödinger equation (NLS) and use these to give a universal blow-up criteria which goes below the scaling invariant threshold. These results are in line with a recent breakthrough construction of finite-time blow-up solutions, and in particular give the first generic result distinguishing potential defocusing blow-up phenomena from many of the known examples of blow-up in the focusing setting. At the end of the talk, we will briefly describe applications to related models.
V-states are uniformly rotating vortex patches of the 2D Euler equation. The only known explicit examples are circles and ellipses, the rest of positive existence results use local or global bifurcation arguments and don’t give any quantitative information of the solutions. I will talk about the existence of solutions far from the perturbative regime, being able to extract nontrivial features of them and a precise quantitative description. We will use a combination of analysis and computer assisted proofs techniques. This is joint work with Javier Gómez-Serrano.
In this talk, we will discuss Leray-Hopf solutions to the two-dimensional Navier-Stokes equations with vanishing viscosity. We aim to demonstrate that when the initial vorticity is only integrable, the Leray-Hopf solutions in the vanishing viscosity limit do not exhibit anomalous dissipation. Moreover, we extend this result to the case where the initial vorticity is merely a Radon measure, assuming its singular part maintains a fixed sign. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. This is a joint work with Luigi De Rosa (University of Basel).
We consider the incompressible Euler equations and related PDEs in scaling critical Sobolev spaces, which are also critical for local well-posedness. We show various ill/well-posedness results for the initial value problem at critical regularity. Then, we discuss some applications of understanding critical dynamics, including singularity formation and enhanced dissipation for the dissipative counterparts.
Suppose u solves the incompressible Navier-Stokes equations on R^d and is bounded by a constant C in some critical space (e.g., L^d) uniformly in time. In this talk we discuss the problem of controlling the solution's regularity explicitly in terms of C, which can shed new light on qualitative regularity theorems such as that of Escauriaza-Seregin-Sverak and potential blow-up phenomena. This work is partly joint with Wojciech Ozanski.
In this talk, I will discuss some optimal boundary regularities for the non cut-off Boltzmann and Landau equations on gerenal bounded domains with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. I also state several open problems in this field.
The Bahouri-Chemin patch is an important singular steady-state for 2D incompressible Euler equations in the flat torus. In this talk, we consider steady states near the Bahouri-Chemin patch. More precisely, we use two different ways to approximate the semi-linear elliptic equations describing the Bahouri-Chemin patch. In one way, we get a smooth curve consisting of smooth steady states close to the Bahouri-Chemin patch in the sense of the Hölder norm of the velocity field. It, in particular, shows the importance of non-vanishing conditions for vorticity on the boundary in the example of double exponential growth for the gradient of vorticity. In the other way, we obtain a continuous curve consisting of singular steady states close to the Bahouri-Chemin patch in certain Hölder norm of the velocity field, which is critical. In the singular steady state examples, an interesting feature is there are multiple particle trajectories across the origin. In the end, if time permits, I will discuss some open questions. This presentation comes from the Joint work of Tarek Elgindi and the Joint work of Chiling Zhang.
It has long been conjectured that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping. We present our recent resolution of this question, uncovering an unexpected contrast between solutions with exponentially-decaying initial data versus those with polynomial decay. Time permitting, we will explain how the proof uses, at a crucial step, results from analytic number theory related to the Riemann zeta function. The talk is based on joint work(s) with Federico Pasqualotto and Yakov Shlapentokh-Rothman.
We prove two results that together strongly suggest that obtaining a positive answer to the Navier-Stokes global regularity question requires more than a refinement of partial regularity theory. First we prove that there exists a wide class of amenable bilinear operators, which contains the Euler operator, whose associated "Navier-Stokes-like" PDE's admit partial regularity in space at the first blowup time. One corollary of this is a partial regularity result for the Navier-Stokes equations in any dimension (without requiring the use of the local energy inequality). Next we prove that there exists an amenable operator whose corresponding "Navier-Stokes-like" PDE admits a solution blowing up in finite time.
This presentation is devoted to studying matrix solutions of the cubic Szegő equation, leading to the following matrix Szegő equation on the 1-d torus and on the real line. The matrix Szegő equation still enjoys a two-Lax-pair structure, which is slightly different from the Lax pair structure of the cubic scalar Szegő equation introduced in Gérard-Grellier [arXiv:0906.4540]. We can establish an explicit formula for general solutions both on the torus and on the real line of the matrix Szegő equation. This presentation is based on the works Sun [arXiv:2309.12136, arXiv:2310.13693].
In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov-Poisson equation, described a phenomenon he called "violent relaxation," a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov-Poisson equation and, in particular, the critical role of regularity of the steady states in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.
The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family as solutions to the vacuum Einstein equations. I will discuss joint work with Yakov Shlapentokh-Rothman (Toronto), where we show that solutions arising from suitably regular initial data decay inverse polynomially in time. Our proof holds for the entire subextremal range of Kerr black hole parameters, (|a| < M).
We develop a new, unified approach to the following two classical questions on elliptic PDE: (i) the strong maximum principle for equations with non-Lipschitz nonlinearities, and (ii) the at most exponential decay of solutions in the whole space or exterior domains (Landis conjecture). Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a weak Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish. If time permits, we will report on some recent C1 estimates with optimized constants and refined Landis-type results. These are based on a new boundary weak Harnack inequality which also has applications in the boundary regularity theory of equations in divergence form.
The control of plasma-wall interaction is one of the keys in a fusion device from both physical and mathematical standpoints. A classical perfect conducting boundary forces the electric field to penetrate inside the domain, which may lead to grazing set singularity in the phase space, preventing the construction of global dynamics for any kinetic PDE plasma models. We establish global asymptotic stability for the relativistic Vlasov-Maxwell-Landau system for describing a collisional plasma specularly reflected at a perfect conducting boundary. This is joint work with Hongjie Dong and Yan Guo.
While the research on water waves modeled by Euler's equations has a long history, mainly in the last two decades traveling periodic rotational waves have been constructed rigorously by means of bifurcation theorems. After introducing the problem, I will present a new reformulation in two dimensions in the pure-gravity case, where the problem is equivalently cast into the form “identity plus compact”, which is amenable to Rabinowitz's global bifurcation theorem. The main advantages (and the novelty) of this new reformulation are that no simplifying restrictions on the geometry of the surface profile and no simplifying assumptions on the vorticity distribution (and thus no assumptions regarding the absence of stagnation points or critical layers) have to be made. Within the scope of this new formulation, global families of solutions, bifurcating from laminar flows with a flat surface, are constructed. Moreover, I will discuss the possible alternatives for the global set of solutions, as well as their nodal properties. This is joint work with Erik Wahlén.
I will discuss a joint work with Jacek Jendrej and Wilhelm Schlag about the two dimensional harmonic map heat flow for maps taking values in the sphere. It has been known since the 90’s that solutions can exhibit bubbling along a well-chosen sequence of times -- the solution decouples into a superposition of concentrating harmonic maps and a body map accounting for the rest of the energy. We prove that every sequence of times contains a subsequence along which such bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubbles in continuous time. The proof uses the notion of “minimal collision energy” developed in the context of the soliton resolution problem for nonlinear waves.
Breathers are solutions of evolutionary PDEs, which are periodic in time and spatially localized. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions (in an infinite dimensional space of periodic in time solutions). In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the origin when the temporal frequency varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. We will obtain an asymptotic formula for the distance between the stable and unstable manifolds when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. This formula allows to say that for a wide set of Klein-Gordon equations breathers do not exist. Due to the exponential small splitting, classical perturbative techniques cannot be applied to this problem.
In this talk, I will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).
Free boundary problems of Bernoulli type arise naturally in fluid dynamics, thermal models, shape optimization, and other contexts. We will focus on the simplest possible archetype problem, and consider many examples of solutions in one and two dimensions. Then we will look at the question of which kinds of solutions are closed under taking limits, and how those limits look like--this is a topic of practical importance for constructing solutions by any argument save for direct minimization. Thanks to a recent breakthrough in joint work with Georg Weiss, we can now give a very precise and descriptive answer to such questions.
We analyze the dynamics of singularities and finite time blowup of generalized Constantin-Lax-Majda equation which corresponds to non-potential effective motion of fluid with competing convection and vorticity stretching terms. Both non-viscous fluid and fluid with various types of dissipation including usual viscosity are considered. An infinite families of exact solutions are found together with the different types of complex singularities approaching the real line in finite times. A nonlinear eigenvalue problem is formulated and solved to determine the rate of blow up and the corresponding self-similar solutions. Both solutions on the real line and periodic solutions are considered. In the periodic geometry, a global-in-time existence of solutions is proven when the data is small and dissipation is strong enough. The found analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for various form of dissipation, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion singularities in the complex plane. The computations validate and extend the analytical theory.
In this talk we will discuss effects of a strong non-favorable constant vorticity on the global dynamics of Stokes waves. We will prove that for constant vorticities above some threshold value the horizontal velocity on the surface is always separated from zero by a constant depending only on the vorticity. In particular this forbids formation of extreme and overhanging waves before a critical layer appears inside the fluid, first at the bottom right below the crest. This also shows that the slopes of unidirectional solutions are uniformly bounded by a small constant, provided the vorticity is large enough and is non-favorable. We also obtain a non-trivial bound for the amplitude. Our analytic results agree with and confirm the numerical analysis by Ko and Strauss (2008). This is a joint work with Miles Wheeler from Bath University, UK.
In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces. I will focus on the case where the spectral window has small width. Indeed the negative curvature assumption leads to improvements over the universal bounds of C.Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. The proof relies on new Strichartz and smoothing estimates for the Schrödinger semi-group, thus illustrating how dispersive PDE techniques can lead to new results in classical harmonic analysis. This is based on joint work with Jean-Philippe Anker and Pierre Germain.
Vortex filaments that evolve according the binormal flow are expected to exhibit turbulent properties. Aiming to quantify this, I will discuss the multifractal properties of the family of functions $R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2},$ which represent the trajectories of regular polygonal vortex filaments and generalize the classical Riemann’s non-differentiable function. I will highlight how the situation seems to critically depend on $x_0$, and I will discuss the important role played by Gauss sums and Diophantine approximation. This talk is based on the recent article in collaboration with Valeria Banica (Sorbonne Université), Andrea Nahmod (University of Massachusetts) and Luis Vega (BCAM, UPV/EHU)
The study of flows over an obstacle is one of the fundamental problems in fluids. In this talk we establish the global validity of the diffusive limit for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior domain. To overcome the well-known difficulty of the lack of Poincare's inequality, we develop a new $L^2-L^6$ splitting for dissipative hydrodynamic part Pf for nonlinear closure.
We will discuss the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p−2}E$ . The electric field may blow up as the distance between insulators approaches zero. We will quantitatively analyze the concentration of the electric field $E$ between the inclusions. This is joint work with Hongjie Dong and Zhuolun Yang.
A fully nonlinear equation is an equation with nonlinearity for the second-order derivatives, which is the highest order. This talk will discuss how such strong nonlinearities affect the regularity of the solution. For the Poisson equation, it is known that the Lp estimate holds for any p > 1. In the case of fully nonlinear equations, Caffarelli showed that the Lp estimate holds for p > n. On the other hand, Pucci obtained an exponent where the Lp estimate fails for fully nonlinear equations. In this talk, we study the Pucci equation, a typical fully nonlinear equation, and show that the Lp estimate holds at p = 1. Note that a special case of the Pucci equation is the Poisson equation, in which case the estimate blows up. I will also prove that there are not only the Pucci's exponent, but also some intervals or more complex sets, where the Lp estimate fails. This talk is based on joint work with Hongjie Dong (Brown University).
How can a fish swim from point A to point B if the ocean current flows faster than the fish can swim? If the current is mean-zero and random in space, then the fish has a chance. I will discuss some results on stochastic homogenization of the G equation, a convex but non-coercive Hamilton-Jacobi equation. Along the way, I will present a simplified proof of a quantitative first-passage percolation shape theorem of Kesten.
We consider nonlocal equations with irregular coefficients and present pointwise gradient estimates in terms of Riesz potentials as well as estimates in terms of certain fractional maximal functions. These pointwise estimates lead to fine higher regularity results in many commonly used function spaces, in the sense that they enable us to detect finer scales that are difficult to reach by more traditional methods. In the case of estimates below the gradient level, we are also able to treat nonlinear equations of fractional p-Laplacian-type. The talk is based on joint works with Lars Diening, Tuomo Kuusi and Yannick Sire.
The existence of a local curve of co-rotating vortex pair solutions to the two-dimensional Euler equations was proven by Hmidi and Mateu via a desingularization of a pair of point vortices. In this talk, we construct a global continuation fo these local curves. The proof relies on an adaptation of the powerful analytic global bifurcation theorem due to Buffoni and Toland, which allows for the singularity at the bifurcation point. Along the global curve of solutions, either the angular fluid velocity vanishes or the two patches self-intersect. This is a joint work with Claudia Garcia (Universidad de Granada).
This talk is about a joint work with Claude Zuily, where we establish several identities related to the equipartition of energy for water waves.
In this talk, I will discuss Sobolev regularity theory for non-local equations on $C^{1,1}$ open sets with both zero and nonzero exterior conditions. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and H\"older estimates of solutions and their arbitrary order derivatives. We use a system of weights consisting of appropriate powers of the distance to the boundary to measure the Sobolev and H\"older regularities of solutions and their arbitrary derivatives. This talk is based on joint works with J.-H. Choi and K.-H. Kim.
We investigate mathematically a plasma boundary layer near the surface of materials immersed in a plasma, called a sheath. From a kinetic point of view, Boyd--Thompson proposed a kinetic Bohm criterion which is required for the formation of sheaths. Then Riemann pointed out (although without a rigorous proof) that the criterion is a necessary condition for the solvability of the stationary Vlasov--Poisson system. Recently, Suzuki--Takayama analyzed rigorously the solvability of the stationary Vlasov--Poisson system, and clarified in all possible cases whether or not there is a stationary solution. It was concluded that the Bohm criterion is necessary but not sufficient for the solvability. In this talk, we study the nonlinear stability and instability of the stationary solutions of the Vlasov--Poisson system. The location of the support of the initial data is a major factor leading to stability/instability. This talk is based on a joint work with Professor M. Takayama (Keio Univ.) and Professor K. Z. Zhang (New York Univ.).
On Corkscrew domains with Ahlfors-regular boundary, we prove the equivalence of the classically considered Lp-solvability of the (homogeneous) Dirichlet problem with the solvability of the inhomogeneous Poisson problem with interior data in an $L_p$-Carleson space (with a natural bound on the $L_p$ norm of the non-tangential maximal function of the solution), and we study several applications. Our main application is towards the $L_q$ Dirichlet-regularity problem for second-order elliptic operators satisfying the Dahlberg-Kenig-Pipher condition (this is, roughly speaking, a Carleson measure condition on the square of the gradient of the coefficients), in the geometric generality of bounded Corkscrew domains with uniformly rectifiable boundaries (although this problem had been open even in the unit ball). Other applications include: several new characterizations of the $L_p$-solvability of the Dirichlet problem, new non-tangential maximal function estimates for the Green's function, a new local T1-type theorem for the $L_p$ solvability of the Dirichlet problem, new estimates for eigenfunctions, free boundary theorems, and a bridge to the theory of the Filoche-Mayboroda landscape function (also known as torsion function). This is joint work with Mihalis Mourgoglou and Xavier Tolsa.
I will discuss the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. We prove that for supercritical surface tension, all known small-amplitude localized waves are (conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shown to be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameter region. The results are obtained by applying a recently developed variant of the Grillakis-Shatah-Strauss method, and the key part of the analysis consists of computing the spectrum of the linearized augmented Hamiltonian at a shear flow or small-amplitude wave. This is joint work with S. Walsh.
The Muskat equation describes the interface of two liquids in a porous medium. We will show that if a solution to the Muskat problem in the case of same viscosity and different densities is sufficiently smooth, then it must be analytic except at the points where a turnover of the fluids happens. We will also show analyticity in a region that degenerates at the turnover points provided some additional conditions are satisfied.
Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In a recent work, we initiated the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians (e.g. on compact quotients of the Heisenberg group). Our results show that nodal sets behave in an anisotropic way which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation and desingularization at singular points. This is a joint work with S. Eswarathasan.
The aim of this talk is to review old and new results concerning the interaction between nonlinearity and weak convergence of pde constrained sequences. This is a ubiquitous theme in the study of nonlinear pde, of which we will place special emphasis on problems with variational structure. We will review classical results in the study of weak (lower semi)continuity of variational integrals, concerning A-quasiconvexity, compensated compactness, and null Lagrangians. We will conclude with new results pertaining primarily to concentration effects in weak convergence, which we used to answer some open questions. Time permitting, we will express our results in the language of generalized Young measures (cf. Di Perna–Majda measures, defect measures). Joint work with A. Guerra, J. Kristensen., and M. Schrecker.
Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative. In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.
I will discuss doubling constructions for minimal surfaces and (depending on time) other recent results and ongoing work. In particular I will discuss recent results in the article by P. McGrath and myself, ``Generalizing the Linearized Doubling approach , I: General theory and new minimal surfaces and self-shrinkers’’,Camb. J. Math. (to appear); arXiv:2001.04240v4’', including a new general area estimate for doublings. If there is time I will briefly discuss some index and characterization results including results in the articles by D. Wiygul and myself ``The index and nullity of the Lawson surfaces $\xi_{g,1}$’’, Camb. J. Math. 8 (2020), 363-405 and ``The Lawson surfaces are determined by their symmetries and topology’’, J. Reine Angew. Math. 786 (2022), 155–173. Finally I will discuss some ongoing work.
In this talk I will discuss the null boundary control of heat-like equations on convex domains, featuring a singular potential that diverges as the inverse square of the distance to the boundary. For this purpose, we will establish global Carleman estimates for the associated operators by combining intermediate inequalities with distinct weights that involve non-smooth powers of the boundary distance. These estimates are sharp in the sense that they capture both the natural boundary conditions and the $H^1$-energy for the problem, which is essential in our controllability argument. Additionally, I will describe the role of the potential strength and the geometry of the domain in our results. This is based on joint work with A. Enciso (ICMAT) and A. Shao (QMUL).
We prove unique continuation properties for linear variable-coefficient Schr\"dinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any cubic exponential rate at two different times must be identically zero. Assuming an extra structural condition, we recover the sharp Gaussian (quadratic exponential) rate uniqueness of free Schr\"odinger flows. Based on joint work with S. Federico (Bologna) and X. Yu (UW).
We will talk about singularity formation for the 3D isentropic compressible Euler and Navier-Stokes equations for ideal gases. These equations describe the motion of a compressible ideal gas, which is characterized by a parameter called adiabatic constant. Finite time singularities for generic adiabatic constants were found in the recent work of Merle, Raphaël, Rodnianski and Szeftel. This is done via a stability analysis around the smooth self-similar profiles for generic adiabatic constants. We will drop the genericity assumption and construct smooth self-similar profiles for all values of the adiabatic constant. In particular, we will construct the first smooth self-similar profile for a monoatomic gas. We also present a different stability analysis around those profiles that allows to show singularity formation for initial data with constant density at infinity. These results are joint work with Tristan Buckmaster and Javier Gomez-Serrano.
We discuss a stabilizing effect of boundary in the context of Vlasov equations: 1) linear Vlasov equation with the diffuse reflection boundary condition, 2) nonlinear Vlasov-Poisson equation with inflow boundary condition.
Our goal is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as semilinear Schrodinger equations or multi-dimensional KdV-type equations. However, our situation here is different since the water waves equations are quasilinear and the solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. I will describe a new strategy to address this fundamental difficulty. This is joint work with Yu Deng and Fabio Pusateri.
We consider the Navier-Stokes equation on a Riemannian manifold. We begin by surveying the results on how the negative curvature of the underlying domain can affect the solutions. We then look into a simple, but fundamental question: what is the ``correct" form of the Navier-Stokes equation on a Riemannian manifold?
Reaction–diffusion equations on Dirichlet domains model populations in territory with hostile boundary. Steady states represent equilibria between reproduction in the interior and mortality at the boundary. On unbounded domains, such steady states are poorly understood even for simple KPP reactions. In this talk, I will show that KPP steady states are unique on domains that satisfy a certain spectral nondegeneracy condition. This is joint work with Henri Berestycki.
We will present recent results on the large particle number and large time effective behavior of conservative or gradient dynamics for particle systems with interactions governed by a Coulomb or more general Riesz potential and subject to possible noise modeling thermal fluctuations. Formally, the empirical measure of such systems converges to a solution of a nonlinear PDE, which is called the mean-field limit, though proving this convergence, how fast it occurs, and how it deteriorates with time is a nontrivial matter. The talk will discuss modulated energy/free energy techniques for studying these questions, in particular new functional inequalities for the variations of these energies that allow to obtain sharp rates of convergence. The talk will also discuss a new method based on proving decay/relaxation estimates for the mean-field PDE that allows for proving rates of convergence which are uniform-in-time. This method is robust and even covers systems with a mild attractive singularity or systems at zero temperature (i.e., without thermal noise).
This talk will consider two aspects of filament hydrodynamics from a PDE perspective. First, we develop a PDE framework for analyzing the error introduced by slender body theory (SBT). SBT facilitates computational simulations of thin filaments in a 3D viscous fluid by approximating the hydrodynamic effect of each fiber as the flow due to a line force density along a 1D curve. Given a 1D force along the fiber centerline, we define a notion of `true' solution to the full 3D slender body problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing. Second, we consider a classical elastohydrodynamic model for the motion of an immersed inextensible filament. We highlight how the analysis can help to better understand undulatory swimming at low Reynolds number. This includes the development of a novel numerical method to simulate inextensible swimmers.
We discuss the unique solvability of parabolic equations in Sobolev spaces when the lower-order coefficients are in mixed $L_{p,q}$ spaces (not necessarily bounded). For the proof, we introduce refined Sobolev embeddings for solutions to parabolic equations in divergence form. We also discuss $L_p$-estimates and solvability results for time fractional parabolic equations with irregular coefficients. The time derivative is the Caputo fractional derivative of order $\alpha \in (0,1)$. This talk is based on joint papers with Seungjin Ryu, Kwan Woo, and Hongjie Dong.
We consider the asymptotic stability of the solitary waves of 1D NLS equations, under the assumption that the linearized operator is generic (no endpoint resonance) and has no internal modes. Moreover, we also consider the 1D nonlinear Klein-Gordon equation with a potential, and give a result on small data existence. The method of analysis is based on the distorted Fourier transform. This is joint work with P. Germain and F. Pusateri.
We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. Freidlin-Wentzell's variational formulas for both a self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the Freidlin-Wentzell's rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.
Rotating stars can be modeled by steady solutions to the Euler-Poisson equations. An extensive literature has established the existence of rotating stars for given differentially rotating angular velocity profiles. However, all of the existing results require the angular velocity to depend on the distance to the rotation axis, but not on the distance to the equatorial plane. Moreover, all of these solutions require a constant entropy within the star. In this talk, I will present a new result which is the first one that allows a general rotation profile, without restrictions. It is also the first result that allows a genuinely changing entropy within the star. This enables us to model the rotation of our own Sun. The addition of entropy causes the old method used to construct such solutions inapplicable. We discover a div-curl reformulation of the problem and perform analysis on the resulting elliptic-hyperbolic system. This is joint work with Juhi Jang and Walter Strauss.
The Peskin problem models the dynamics of a closed elastic string immersed in an incompressible 2D stokes fluid. This set of equations was proposed as a simplified model to study blood flow through heart valves. The immersed boundary formulation of this problem has proven very useful in particular giving rise to the immersed boundary method in numerical analysis. In a joint work with Stephen Cameron, we consider the general case of a fully non-linear tension law. We prove local wellposedness for arbitrary initial data in the scaling critical Besov space $\dot{B}^{3/2}_{2,1}$, and the high order smoothing effects for the solution.
I will talk about the new construction of genus-zero free boundary minimal surfaces embedded in the unit ball in the Euclidean three-space which are compact and lie arbitrarily close to the boundary unit sphere with an arbitrarily large number of connected boundary components. The construction is by PDE gluing methods and the surfaces are desingularizations of unions of many catenoidal annuli and two flat discs. The talk is based on the joint walk with Professor Nicolaos Kapouleas.
The Vlasov-Poisson system is a kinetic model for a continuous density of particles interacting through either Newtonian or Coulombic gravitation. I will describe the scattering problem for this equation, where one must find the asymptotic dynamic as time goes to infinity, and then connect the asymptotic behavior of the solution at time minus infinity to time plus infinity through the so-called scattering map. This model exhibits "modified" scattering, where the asymptotic dynamic is given by the linearized equation, plus an explicit nonlinear correction. To solve the scattering problem, we apply the pseudo-conformal transformation, more widely used in the study of the nonlinear Schrodinger equation. This transformation, which inverts time, allows us to reformulate the scattering problem as a Cauchy problem, which we then solve using Picard iteration. This talk is based on joint work with Benoit Pausader, Zhimeng Ouyang, and Klaus Widmayer.
The talk is to precise the classical notion of Landau damping in a collisionless plasma, to give a complete overview of the linear damping theory, and to present progress on the nonlinear damping problem.
In this talk we will discuss some recent local wellposedness results for the free boundary incompressible Euler equations. We will consider a functional setting in which the interfaces can exhibit corners and cusps. Contrary to what happens in all the previously known non-C^1 water waves, the angle of these crests can change in time. The talk is based on joint work with Diego Cordoba and Nastasia Grubic.
We consider the 2D gravity water waves equation on an infinite domain. We prove that the time of existence of solutions is uniform even as the gravity parameter $g \to 0$. The energy estimate used to prove this is moreover scaling invariant with respect to all the scaling transformations of the Euler equation. The energy also allows for interfaces with angled crests and cusps and we also prove an existence result for $g = 0$. As an application of this energy estimate, we then consider the water wave equation with no gravity where the fluid domain is homeomorphic to the disc. We prove a local wellposedness result which allows for interfaces with angled crests and cusps and then by a rigidity argument, we show that there exists initial interfaces with angled crests for which the energy blows up in finite time, thereby proving the optimality of this local wellposedness result.
In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock-forming solutions to the multi-dimensional relativistic and non-relativistic compressible Euler equations. A lot of the progress has relied on geometric techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress, with a focus on results in 3D that reveal various aspects of the structure of the maximal (classical) development of the data. Roughly speaking, the maximal development is the largest possible classical solution + region that is uniquely determined by smooth initial data. I will also discuss the implications of the structure of the maximal development for the shock development problem, which is the problem of continuing the solution weakly after the onset of the shock by describing the transition of the solution from classical to weak. Finally, I will describe some open problems and speculate on the future of the field. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.
In this talk, I will introduce some recent results regarding equations with non-local operators. In particular, we focus on a weighted mixed-norm estimate for fractional wave equations and Sobolev estimates for fractional parabolic equations with space-time non-local operators. This is a joint work with Hongjie Dong.
This talk will present the soliton resolution for the radial quadratic semilinear wave equation in six dimensions. In a joint work with T. Duyckaerts, C. Kenig, and F. Merle, we prove that any spherically symmetric solution, that remains bounded in the energy norm, evolves asymptotically to a sum of decoupled stationary states, plus a radiation term. As a by-product of the approach we prove the non-existence of multisoliton solutions that do not emit any radiation. The proof follows the method initiated for large odd dimensions by Duyckaerts, Kenig and Merle, reducing the problem to ruling out the existence of such non-radiative multisolitons, by deriving a contradiction from a finite dimensional system of ordinary differential equations governing their modulation parameters. In comparison, the difficulty in six dimensions is the failure of certain channel of energy estimates and the related existence of a linear resonance. The main novelties are the obtention of new channel of energy estimates, as well as the classification of non-radiative solutions with small energy.
Recently, the modulated self-similarity technique has achieved success in fluid dynamic equations. In this talk, we apply this technique to establish finite time shock formation of the Burgers-Hilbert equation. The shocks are asymptotic self-similar at one single point. The shocks can be stable or unstable, both of which have an explicitly computable singularity profile, and the shock formation time and location are described by explicit ODEs. Both cases utilize a transformation to appropriated self-similar coordinates, the quantitative properties of the corresponding self-similar solution to the inviscid Burgers' equation, and transport estimates. In the case of unstable shock, we, in addition, control the two unstable directions by Newton's iteration.
In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.
The Einstein--massless Vlasov system is a relevant model in the study of collisionless many particle systems in general relativity. In this talk, I will present a stability result for the exterior of Schwarzschild as a solution of this system assuming spherical symmetry. We exploit the hyperbolicity of the geodesic flow around the black hole to obtain decay of the energy momentum tensor, despite the presence of trapped null geodesics. The main result requires a precise understanding of radial derivatives of the energy momentum tensor, which we estimate using Jacobi fields on the tangent bundle in terms of the Sasaki metric.
Whitham's equation is a nonlinear, nonlocal, very weakly dispersive shallow water wave model in one space dimension. In this talk we are concerned with non-smooth traveling wave solutions of extreme form to this equation. Their existence was conjectured by Whitham in 1967 and established by Ehrnström and Wahlén just a few years ago, who proved that there is a monotone traveling wave of sharp $C^{1/2}$-Hölder regularity. I will show that there is only one monotone, even, periodic traveling wave of greatest height featuring a cusp at the origin and that, as widely believed in the community, its profile is in fact convex between crest and trough. This can be understood as the counterpart, in the case of the Whitham equation, of the landmark results on the uniqueness and convexity of Stokes wave for Euler. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.
For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are Holder continuous. We construct time-independent supercritical drifts in $L^{n−\lambda}(R^n)$ with arbitrarily small $\lambda>0$ such that the Harnack inequality and the Holder continuity fail in both the elliptic and the parabolic cases, in dimensions $n \geq 3$. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimension 3.
The Benjamin--Ono (BO) equation, which describes internal long waves of deep stratified fluids, has multi-soliton solutions. I shall prove the invariance of the multi-soliton manifold $\mathcal{U}_N$, given by \begin{equation*} \mathcal{U}_N:=\{u \in L^2(\mathbb{R}, \mathbb{R}) : u(x)=\sum_{j=1}^N \frac{2\eta_j}{ (x-x_j)^2 + \eta_j^2 } , \quad \eta_j >0, \quad x_j \in \mathbb{R}, \quad \forall 1\leq j\leq N\}, \quad \forall N \geq 1, \end{equation*}under the BO flow and construct global (generalized) action--angle coordinates of the BO equation on $\mathcal{U}_N$ in order to solve this equation by quadrature for any initial datum $u_0 \in \mathcal{U}_N$. The complete integrability of the BO equation on every $\mathcal{U}_N$ constitutes a first step towards the soliton resolution conjecture of the BO equation on the line. The construction of such coordinates relies on the Lax pair structure of the BO equation, the inverse spectral transform and the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an $N$-soliton $u \in \mathcal{U}_N$ provides a spectral connection between the Lax operator and the infinitesimal generator of the shift semigroup acting on some Hardy spaces. Furthermore, $\mathcal{U}_N$ can be interpreted as the universal covering of the manifold of $N$-gap potentials $U_N^{\mathbb{T}}$ for the space-periodic BO equation as described by Gérard--Kappeler
It is well-known that in three space dimensions, smooth solutions to the equations describing a compressible gas can break down in finite time. One type of singularity which can arise is known as a "shock", which is a hypersurface of discontinuity across which the integral forms of conservation of mass and momentum hold and through which there is nonzero mass flux. One can find approximate solutions to the equations of motion which describe expanding spherical shocks. We use these model solutions to construct global-in-time solutions to the irrotational compressible Euler equations with shocks. This is joint work with Igor Rodnianski.
The sine-Gordon model is a classical nonlinear scalar field theory that was discovered in the 1860s in the context of the study of surfaces with constant negative curvature. Its equation of motion features soliton solutions called kinks and breathers, which play an important role for the long-time dynamics. I will begin the talk with an introduction to classical 1D scalar field theories and the asymptotic stability problem for kinks. After surveying recent progress on the problem, I will present a joint work with W. Schlag on the asymptotic stability of the sine-Gordon kink under odd perturbations. Our proof is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key aspects are a super-symmetric factorization property of the linearized operator and a remarkable non-resonance property of a variable coefficient quadratic nonlinearity.
In this talk I will review the classical Cauchy problem for Einstein equations. I will explain some of its geometric features and recast the equations as a system of coupled quasilinear transport-elliptic-Maxwell equations. I will present the global-in-time existence conjecture (aka the conjecture of weak cosmic censorship) and how low regularity local existence results (as the celebrated bounded $L_2$ curvature theorem) can be used to get insight on the formation of singularities. I will then review the classical bounded $L_2$ curvature theorem of Klainerman-Rodnianski-Szeftel and present a version generalised to initial data posed on an initial spacelike and an initial characteristic hypersurface that I obtained jointly with Stefan Czimek.
Fusion energy is at the threshold of becoming one of the most green and sustainable energy sources in the world. This energy is created by heating an ionized gas (plasma) to extreme temperatures in order to allow high energy particle collisions to occur. This leads to an exothermic fusion reaction releasing immense energy to be harvested. One major hurdle, is the plasma is highly pressurized and must be contained within a reactor. A solution to this issue is applying a strong magnetic field which traps the particles from escaping radially outwards from the confinement chamber. Such a system can be modeled mathematically by the Hot, Magnetized Relativistic Vlasov Maxwell (HMRVM) system. A small physically pertinent parameter $\epsilon$, with $0 < \epsilon \ll 1$, related to the inverse of a gyrofrequency, governs the strength of a spatially inhomogeneous applied magnetic field given by the function $x \to \epsilon^{-1} \mathbf{B}_e (x)$. Stationary (equilibrium) solutions to this system are well understood, but it is not clear how perturbations from equilibrium could lead to destabilization of the plasma (the plasma explodes releasing uncontrollable energy). It has been recently in shown that, in the case of neutral, cold, and dilute plasmas (like in the Earth's magnetosphere), smooth solutions corresponding to perturbations of equilibria exist on a uniform time interval $[0, T]$, with $0 < T$ independent of $\epsilon$. In this talk we further extend these results to hot plasmas for well prepared initial data. This is a joint work with D. Preissl and C. Cheverrey.
I will report on progress obtained for the $W^{s,p}$-regularity theory for nonlocal/fractional equations of differential order 2s with bounded measurable Kernel. Namely, under (not yet optimal) assumptions on the kernel we obtain $W^{t,p}$-estimates for suitable right-hand sides, where $s< t < 2s$. Technically we compare such equations via a commutator estimate to a simpler fractional equation. Based on joint works with M. Fall, T. Mengesha, S. Yeepo.
We present some results concerning the solvability of linear elliptic equations in bounded domains with the main coefficients almost in VMO, the drift and the free terms in Morrey classes containing $L_{d}$, and bounded zeroth order coefficient. We prove that the second-order derivatives of solutions are in a local Morrey class containing $W^{2}_{p,\text{loc}}$. Actually, the exposition is given for fully nonlinear equations and encompasses the above mentioned results, which are new even if the main part of the equation is just the Laplacian.
In this talk, we will consider the low regularity well-posedness problem for the two dimensional gravity water waves. This quasilinear dispersive system admits an interesting structure which we exploit to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier energy estimates of Hunter-Ifrim-Tataru. These results allow us to significantly lower the regularity threshold for local well-posedness, even without using dispersive properties. Combined with nonlinear vector field Sobolev inequalities, an idea first introduced by the last two authors in the context of the Benjamin-Ono equations, these improvements extend to global solutions for small and localized data. This is joint work with Mihaela Ifrim and Daniel Tataru.
In this talk I will present an existence result for two-dimensional steady solitary water waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. The particularity of these waves is that they may have internal stagnation points and overhanging wave profiles. The proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain: one describing the conformal map of the fluid region and the other the flow beneath the wave. This is a joint work with Miles. H. Wheeler (University of Bath).
The mathematically rigorous derivation of a nonlinear Boltzmann equation from first principles is an extremely active research area. In classical physical systems, this has been achieved in various models, based on a variety of fundamental works. In the quantum case, the problem has essentially remained open. I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics of a Bose-Einstein condensate, starting with the von Neumann equation for an interacting Boson gas. This is based on joint work with Thomas Chen.
In this talk, we will talk about our recent results on the traveling wave solution to the Burgers-Hilbert equation, specifically their long-term stability under small perturbations of the initial data, with a lifespan beyond the reach of local wellposedness. This is joint work with Castro and Cordoba.
We discuss the insulated conductivity problem with multiple inclusions embedded in a bounded domain in n-dimensional Euclidean space. The gradient of a solution may blow up as two inclusions approach each other. The optimal blow-up rate was known in dimension $n=2$. It was not known whether the established upper bound of the blow-up rates in higher dimensions were optimal. We answer this question by improving the previously known upper bound of the blow-up rates in dimensions $n \geq 2$. This is a joint work with Yanyan Li.
We investigate behavior of solutions to the generalized Korteweg-de-Vries (KdV) and Benjamin-Ono (BO) equations from the numerical point of view. The efficient (FFT accessible) and energy-conservative spatial discretization (either Fourier basis functions for fast decay solutions or Wiener rational basis functions for slow decay solutions) are used. The energy conservative schemes can be constructed with a variety of the energy-conservative time integrators, and eventually applied for our numerical study. Our numerical simulations justify the soliton resolution conjecture for the $L^2$-subcritical cases. Furthermore, we study threshold criteria for the global/local existence of solutions in critical and supercritical cases. We also study the dynamics and the structure of the singular (blow-up) solutions, which have some similarities to the blow-up in the nonlinear Schrodinger equation, i.e., slow convergence to the ground state profile solution to the $L^2$-critical case, and fast convergence to the corresponding profile in the $L^2$-supercritical cases. Finally, we investigate their higher-dimensional generalizations, namely, Zakharov-Kuznetsov (ZK) and Higher dimensional BO (HBO) equations, respectively. We discuss the asymptotic stability in the 3d ZK equation and the existence of finite time blow-up in the 2d HBO equation.
In this talk we will highlight recent discoveries in kinetic theory from two recent papers. We will develop an uncertainty principle, a new a-priori bound and the concept of a blind cone with respect to an observer for the evolution particles in the mesoscopic scale solely based on conservation laws. We will show that the energy within any bounded set of the spatial variable is integrable over time and that the total mass of the particles concentrates within a specific collection of arbitrarily acute blind cones with respect to any observer. This shows that, as uncertainty inevitably increases, particles will move away in a radial manner from any fixed observer thereby erasing the angular component of momentum. We will also discuss a generalization of these results for the interactions instead of the particles and establish analogies to Morawetz and interaction Morawetz estimates for the nonlinear Schrodinger equation. These results are independent of the specific structure of interactions, therefore they are also true for the Boltzmann equation. We will end the talk with results about the case of the Boltzmann equation. We will introduce the concept of a scattering frame of reference and show the existence and uniqueness of a specific class of classical solutions to the Boltzmann equation. These solutions scatter to linear states in the $L^{\infty}$ norm. Furthermore, we will discuss the asymptotic completeness of this class of solutions and establish another connection with the case of the nonlinear Schrodinger equation. Notably, this shows that solutions of the Boltzmann equation do not necessarily converge to a Maxwellian but can scatter to linear states arbitrarily close to any prescribed linear state.
We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three-dimensional incompressible Euler equations. The flow is axi-symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. It is a solution minimizing the kinetic energy under natural constraints on vortex impulse, mass and strength so that its Lyapunov stability follows from a variational method. We also visit stability results of other vortex solutions in the plane including circular vortex patches, Lamb's dipoles. This talk is based on joint works with K. Abe(Osaka City Univ.) and with D. Lim(UNIST).
In this talk, I will present local/global well-posedness results for the 2d/3d Muskat problem in critical spaces. Moreover, I will also discuss a toy model for the 2d Muskat which is globally well-posed for large data in a supercritical space. This is based on works in collaboration with Thomas Alazard, Omar Lazar, Ke Chen and Yiran Xu.
We consider the Kuramoto-Sivashinsky equation (KSE) on the two-dimensional torus in scalar form. We prove global existence for small data in the absence of growing modes. If growing modes are present, we show that global existence for arbitrary data holds for the advective KSE, provided the advecting flow field induces a sufficient small diffusion time for the linearized operator, for example if the flow is mixing with large amplitude. If the advecting flow is a shear flow, then we show global existence still holds by using pseudo-spectral estimates.
I will discuss recent developments in our study of hydrodynamic swarming governed by general systems of alignment hydrodynamics.
A main question of interest is to characterize the emergent behavior of such systems, which we quantify in terms of the spectral gap of a weighted Laplacian associated with the alignment operator.
Our spectral analysis does not require thermal equilibrium (no closure for the pressure), covers both long-range and short-range kernels, and yields, in the canonical case of metric-based short-range kernels, an emergent behavior of non-vacuous smooth solutions.
Global smooth solutions are known to exist in mono-kinetic closure in one and two spatial dimensions, subject to sub-critical initial data. We settle the question for arbitrary dimension, obtaining non-trivial initial threshold conditions which guarantee existence of multiD global smooth solutions.
We are concerned with the question of well-posedness of stochastic three dimensional incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak--strong uniqueness; (iii) non-uniqueness in law; (iv) existence of a strong Markov solution; (v) non-uniqueness of strong Markov solutions; all hold true within this class. Moreover, as a byproduct of (iii) we obtain existence and non-uniqueness of probabilistically strong and analytically weak solutions defined up to a stopping time and satisfying an energy inequality.
This talk is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical interest in this system, the prior work on this problem is limited to Lagrangian coordinates, in high regularity spaces. Instead, the objective of the present work is to provide a new, fully Eulerian approach to this problem, which provides a complete, Hadamard style well-posedness theory for this problem in low regularity Sobolev spaces. In particular we give new proofs for both existence, uniqueness, and continuous dependence on the data with sharp, scale invariant energy estimates, and continuation criterion. This is joint work with Daniel Tataru
A celebrated theorem of Jorgens-Calabi-Pogorelov says that global convex solutions to the Monge-Ampere equation $\det(D^2u) = 1$ are quadratic polynomials. On the other hand, an example of Pogorelov shows that local solutions can have line singularities. It is natural to ask what kinds of singular structures can appear in functions that solve the Monge-Ampere equation outside of a small set. We will discuss examples of functions that solve the equation away from finitely many points but exhibit polyhedral and Y-shaped singularities. Along the way we will discuss geometric and applied motivations for constructing such examples, as well as their connection to a certain obstacle problem for the Monge-Ampere equation.
Gaussian free field is a generalization of the standard Brownian motion and turns out to be the stationary solution of the heat equation with additive space-time white noise. In the whole space, the analysis leads to other types of Gaussian fields, as well as interesting phenomena in dimensions one and two.
In this talk, I will discuss several regularity criteria that provide geometric constraints on the possible finite-time blowup of solutions of the Navier-Stokes equation. This approach, based on the strain formulation of the Navier-Stokes regularity problem, not only gives geometric criteria for blowup in terms of the eigenvalues of the strain matrix, but also improves previous geometric regularity criteria involving the vorticity. Finally, I will discuss finite-time blowup for a model equation for the self-amplification of strain that respects these geometric constraints.
A Stokes wave, namely a $2\pi/\kappa$-periodic nonlinear surface wave on an incompressible inviscid fluid of unit depth, is known to be subject to the Benjamin-Feir modulational instability when the wave number $\kappa$ satisfies $1.3627\ldots<\kappa$. The Benjamin-Feir instability was proved by Bridges and Mielke in 1995. Besides the instability of Benjamin-Feir’s, numerical investigations suggested that the Stokes wave may be subject to some additional instabilities. In particular, recent work of Deconinck and Oliveras found bubbles of unstable spectra near the imaginary axis which they referred to as ''high-frequency instabilities''. However, there was no rigorous treatment for the additional instabilities. We reformulated, by using flattening coordinates, the Euler equations to equations on an infinite cylindrical domain. The Stokes expansion for the Stokes wave was then calculated in the new coordinates. To study the spectrum of the Stokes wave of small amplitude, we reduced the eigenvalue system, by a center manifold reduction method, to a dynamical system on a finite-dimensional reduced space. We then defined, for the first time, Gardner's periodic Evans function associated with the Stokes wave of small amplitude. By analyzing the periodic Evans function near the origin, we recovered the well-known Benjamin-Feir instability, giving an alternative proof. Our method can also study the spectrum away from the origin. In particular, we proved that the wave is unstable at the ``resonance high-frequency of order 2” provided that $0.86430<\kappa<1.00804$, justifying the existence of additional instabilities
For self-adjoint pseudodifferential operators of order 0, Colin de Verdiere and Saint-Raymond introduced natural dynamical conditions (motivated by the study of internal waves in fluids) guaranteeing absolute continuity of the spectrum. I will present an alternative approach to obtaining such results based on Melrose’s radial propagation estimates from scattering theory (joint work with S. Dyatlov). I will then explain how an adaptation of the Helffer–Sjöstrand theory of scattering resonances shows that in a complex neighbourhood of the continuous spectrum viscosity eigenvalues have limits as viscosity goes to 0. Here the viscosity eigenvalues are the eigenvalues of the original operator to which an anti-self-adjoint elliptic 2nd order operator is added. This justifies claims made in the physics literature (joint work with J Galkowski).
We are concerned with interior and global pointwise gradient estimates for solutions to singular quasilinear elliptic equations with measure data, whose prototype is given by the $p$-Laplace equation $-\Delta_p u=\mu$ with $p\in (1,2)$. Interior and global modulus of continuity estimates of the gradients of solutions are also established. This is a joint work with Hongjie Dong.
A classical problem in the Calculus of Variations asks to find a curve enclosing a region with maximum area.
In this talk I shall discuss the seemingly opposite problem of finding curves enclosing a region with MINIMUM area. As a motivation, one may think of a forest fire, where firemen seek to construct a barrier, minimizing the region burned by the fire.
In this model, a key parameter is the speed at which the barrier is constructed. If the construction rate is too slow, the fire cannot be contained.
The talk will focus on two main questions:
Based on the analysis of a corresponding Hamilton-Jacobi equation with obstacles, results on the existence or non-existence of a blocking strategy will be presented, together with a new regularity result for optimal barriers, and an example where the optimal barrier can be explicitly computed.
I will describe recent progress on the singularity problem for solutions to the 3D Euler equation.
We develop the $\ell_p$-theory of space-time stochastic difference equations which can be considered as a discrete counterpart of N.V. Krylov's $L_p$-theory of stochastic partial differential equations. We also prove a Calderon-Zygmund type estimate for deterministic parabolic finite difference schemes with variable coefficients under relaxed assumptions on the coefficients and the forcing term.
We consider the relativistic Euler equations with a physical vacuum boundary and an equation of state $p(\varrho)=\varrho^\gamma$, $\gamma > 1$. We establish the following results. (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the ow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in $L^1_t Lip_x$ and a suitable weighted version of the density is at the same regularity level. This is joint work with Mihaela Ifrim and Daniel Tataru.
Consider the effective Hamiltonian $\bar H(p)$ associated with the mechanical Hamiltonian $H(p,x)={1\over 2}|p|^2+V(x)$. One of the major open problems in the homgenization theory is to identify the shape of effective Hamiltonians. In this talk, I will present a recent progress that says , for generic V, the effective Hamiltonian is piecewise 1d in a dense open set in two dimensions although the original $H(p,x)$ is quadratic on $p$. A brief introduction of the main tool of the proof --Aubry-Mather theory--will be provided as well.
I will describe some recent results about the validity of the Smoluchowski-Kramers approximation for a class of stochastically forced damped wave equations in case the friction coefficient depends on the status.
We consider the Vlasov-Poisson-Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near the Maxwellian equilibrium state with decay in time and some regularity results for small initial perturbations, in any general bounded domain (including a torus as in a tokamak device), in the presence of specular-reflection boundary condition. We provide a new improved $L^{2}\rightarrow L^{\infty}$ framework: $L^{2}$ energy estimate combines only with $S_{\mathcal{L}}^{p}$ estimate for the ultra-parabolic equation. This is a joint work with Hongjie Dong and Yan Guo.
We exhibit a large family of new, non-trivial stationary states of square torus. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow. This is in contrast with both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel, for which we can show that the only stationary states near them must be shears. This has surprising consequences in the context of inviscid damping in 2D Euler and enhanced dissipation in Navier-Stokes.
We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller-Segel system in the subcritical regimes, is obtained.
We consider a system of $N$ particles evolving according to the gradient flow of their Coulomb or Riesz interaction, or a similar conservative flow, and possible added random diffusion. By Riesz interaction, we mean inverse power $s$ of the distance with $s$ between $d-2$ and $d$ where $d$ denotes the dimension. We present a convergence result as $N$ tends to infinity to the expected limiting mean field evolution equation. We also discuss the derivation of Vlasov-Poisson from newtonian dynamics in the monokinetic case, as well as related results for Ginzburg-Landau vortex dynamics.
In 1959 George Batchelor predicted that a passively advected quantity in a fluid (e.g. temperature or some chemical concentration), in a regime where the scalar dissipation is much lower than the fluid viscosity, should display a power spectral density that behaves like $1/|k|$ over an appropriate inertial range, known as "the Batchelor spectrum" (or Batchelor's law). Since then this prediction has been observed experimentally (for instance in salinity variations in the upper ocean) and in various numerical experiments.
In this talk I will discuss a recent result that rigorously proves Batchelor's law for the cumulative power spectrum of a scalar undergoing advection diffusion by the incompressible stochastic Navier-Stokes equations (among a host of other stochastic fluid models) in $\mathbb{T}^2$. The proof relies on an analysis of the chaotic properties of the associated Lagrangian flow and a quantification of the almost sure mixing rate of this flow uniformly in the diffusivity. In particular, we will see how exponential decay of the scalar in $H^{-1}$ follows from a recent proof of a positive Lyapunov exponent for the Lagrangian flow and a careful spectral analysis of an associated Feynmann-Kac semi-group associated to the evolution of two-point Lagrangian statistics.
This is joint work with Jacob Bedrossian and Alex Blumenthal
All the elastic materials are more or less compressible in practice. This talk is concerned about the large-scale regularity for the linear system of nearly incompressible elasticity. In the setting of stochastic homogenization, we develop some new techniques to establish the large-scale estimates of the displacement and the pressure, which are uniform in both the scale parameter and the incompressibility parameter, in non-smooth domains whose boundaries are smooth at large scales and bumpy at small scales.
In this talk, we will discuss recent results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer and Buffoni--Toland, but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs.
As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.
This is joint work with Ming Chen and Miles H. Wheeler.