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.