Brown PDE Seminar

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:

• Can the fire be confined within a bounded region?
• If so, is there an optimal strategy for constructing the barrier, minimizing the total value of the land destroyed by the fire?

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.