Brown PDE Seminar

In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex. This is based on an upcoming joint work with Jonathan Luk at Stanford.

We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single "eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. We will show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. This is joint work with S. Iyer and T. Nguyen.

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 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.

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).