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Series: PDE Seminar

I will discuss applications of the theory of gradient ﬂows to the dynamics of evolution equations. First, I will review how to obtain convergence rates towards equilibrium in the strictly convex case. Second, I will introduce a technique developed in collaboration with Moon-Jin Kang that allows one to obtain convergence rates towards equilibrium in some situations where convexity is not available. Finally, I will describe how these techniques were useful in the study of the dynamics of homogeneous Vicsek model and the Kuramoto-Sakaguchi equation. The contributions on the Kuramoto-Sakaguchi equation are based on a joint work with Seung-Yeal Ha, Young-Heon Kim, and Jinyeong Park. The contributions to the Vicsek model are based on works in collaboration with Alessio Figalli and Moon-Jin Kang.

Series: PDE Seminar

We consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions. We introduce a regularized problem that preserves the monotonicity and prove the existence of solutions to the regularized problem. Next, using Minty's method, we establish the existence of solutions for the original MFGs. Finally, we examine the properties of these weak solutions in several examples.

Series: PDE Seminar

Linear wave solutions to the Charney-Hasegawa-Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface. We give a rationalparametrization of the smooth points on this surface, answering the question: What are all resonant triads, and how may they be enumerated quickly? We also give a fiberwise description, yielding an algorithmic procedure to answer the question: For fixed $r \in \Q$, what are all wavevectors $(x,y)$ that resonate with a wavevector $(a,b)$ with $a/b = r$?

Series: PDE Seminar

Abstract: Abstract: Let p(x,xi) be a complex-valued polynomial of degree 2 on R^{2n}, and let P be the corresponding non-self-adjoint Weyl quantization. We will discuss some results on the relationship between the classical Hamilton flow exp(H_p) and the L^2 operator theory for the Schrödinger evolution exp(-iP), under a positivity condition of Melin and Sjöstrand.

Series: PDE Seminar

We study the quasilinear wave equation $\Box_{g} u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

Series: PDE Seminar

This talk is about the Dirac equation. We consider an electron modeled by awave function and evolving in the Coulomb field generated by a nucleus. Ina very rough way, this should be an equation of the form$$i\partial_t u = -\Delta u + V( \cdot - q(t)) u$$where $u$ represents the electron while $q(t)$ is the position of thenucleus. When one considers relativitic corrections on the dynamics of anelectron, one should replace the Laplacian in the equation by the Diracoperator. Because of limiting processes in the chemistry model from whichthis is derived, there is also a cubic term in $u$ as a correction in theequation. What is more, the position of the nucleus is also influenced bythe dynamics of the electron. Therefore, this equation should be coupledwith an equation on $q$ depending on $u$.I will present this model and give the first properties of the equation.Then, I will explain why it is well-posed on $H^2$ with a time of existencedepending only on the $H^1$ norm of the initial datum for $u$ and on theinitial datum for $q$. The linear analysis, namely the properties of thepropagator of the equation $i\partial_t u = D u + V( \cdot - q(t))$ where$D$ is the Dirac operator is based on works by Kato, while the non linearanalysis is based on a work by Cancès and Lebris.It is possible to have more than one nucleus. I will explain why.(Joint work with F. Cacciafesta, D. Noja and E. Séré)

Series: PDE Seminar

A major open problem in periodic homogenization of Hamilton-Jacobi equations is to understand deep properties of the effective Hamiltonian. In this talk, I will present some related works in both convex and non-convex situations. If time permits, relevant problems from applications in turbulent combustion and traffic flow will also be discussed.

Series: PDE Seminar

We investigate Lipschitz maps, I, mapping $C^2(D) \to C(D)$, where $D$ is an appropriate domain. The global comparison principle (GCP) simply states that whenever two functions are ordered in D and touch at a point, i.e. $u(x)\leq v(x)$ for all $x$ and $u(z)=v(z)$ for some $z \in D$, then also the mapping I has the same order, i.e. $I(u,z)\leq I(v,z)$. It has been known since the 1960’s, by Courr\`{e}ge, that if I is a linear mapping with the GCP, then I must be represented as a linear drift-jump-diffusion operator that may have both local and integro-differential parts. It has also long been known and utilized that when I is both local and Lipschitz it will be a min-min over linear and local drift-diffusion operators, with zero nonlocal part. In this talk we discuss some recent work that bridges the gap between these situations to cover the nonlinear and nonlocal setting for the map, I. These results open up the possibility to study Dirichlet-to-Neumann mappings for fully nonlinear equations as integro-differential operators on the boundary. This is joint work with Nestor Guillen.

Series: PDE Seminar

The incompressible three-dimensional Euler equations are a basic model of fluid mechanics. Although these equations are more than 200 years old, many fundamental questions remain unanswered, most notably if smooth solutions can form singularities in finite time. In this talk, I discuss recent progress towards proving a finite time blowup for the Euler equations, inspired numerical work by T. Hou and G. Luo and analytical results by A. Kiselev and V. Sverak. My main focus lies on various model equations of fluid mechanics that isolate and capture possible mechanisms for singularity formation. An important theme is to achieve finite-time blowup in a controlled manner using the hyperbolic flow scenario in one and two space dimensions. This talk is based on joint work with B. Orcan-Ekmecki, M. Radosz, and H. Yang.

Series: PDE Seminar

SGSW is a third level specialization of Navier-Stokes (via
Boussinesq, then Semi-Geostrophic), and it accurately describes
large-scale, rotation-dominated atmospheric flow under the extra-assumption
that the horizontal velocity of the fluid is independent of the vertical
coordinate. The Cullen-Purser stability condition establishes a connection
between SGSW and Optimal Transport by imposing semi-convexity on the
pressure; this has led to results of existence of solutions in dual space
(i.e., where the problem is transformed under a non-smooth change of
variables). In this talk I will present recent results on existence and weak
stability of solutions in physical space (i.e., in the original variables)
for general initial data, the very first of their kind. This is based on
joint work with M. Feldman (UW-Madison).