Differential Equations

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Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy. C. De Lellis and L. Székelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration. This approach involves correcting “approximate solutions" by adding rapid oscillations, which are designed to reduce the error term in solving the equation. In this talk, I will discuss an
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The main focus of this talk is a class of asymptotic methods called averaging. These methods approximate complicated differential equations that contain multiple scales by much simpler equations. Such approximations oftentimes facilitate both analysis and computation. The discussion will be motivated by simple examples such as bridge and swing, and it will remain intuitive rather than fully rigorous. If time permits, I will also mention some related projects of mine, possibly including circuits, molecules, and planets.
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We survey some recent results by the speaker, Jason Metcalfe and Daniel Tataru for small data local well-posedness of quasilinear Schrödinger equations. In addition, we will discuss some applications recently explored with Jianfeng Lu and recent progress towards the large data short time problem. Along the way, we will attempt to motivate analysis of the problem with connections to problems from Density Functional Theory.
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A class of kinetic models for the collective self-organization of agents is presented. Results on the global existence of weak solutions as well as a hydrodynamic limit will be discussed. The main tools employed in the analysis are the velocity averaging lemma and the relative entropy method. This is joint work with T. Karper and A. Mellet.
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The existence of large BV (total variation) solution for compressible Euler equations in one space dimension is a major open problem in the hyperbolic conservation laws, where the small BV existence was first established by James Glimm in his celebrated paper in 1964. In this talk, I will discuss the recent progress toward this longstanding open problem joint with my collaborators. The singularity (shock) formation and behaviors of large data solutions will also be discussed.
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Abstract: the theory of weak turbulence has been put forward by appliedmathematicians to describe the asymptotic behavior of NLS set on a compactdomain - as well as many other infinite dimensional Hamiltonian systems.It is believed to be valid in a statistical sense, in the weaklynonlinear, infinite volume limit. I will present how these limits can betaken rigorously, and give rise to new equations.
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Shock waves are idealizations of steep spatial gradients of physical quantities such as pressure and density in a gas, or stress in an elastic solid. In this talk, I outline the mathematics of shock waves for nonlinear partial differential equations that are simple models of physical systems. I will focus on non-classical shocks and smooth waves that they approximate. Of particular interest are comparisons between nonlinear traveling waves influenced strongly by dissipative effects such as viscosity or
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For a $C^{1,1}$-uniformly elliptic matrix $A$, let $H(x,p)=$ be the corresponding Hamiltonian function. Consider the Aronsson equation associated with $H$: $$(H(x,Du))x H_p(x,Du)=0.$$ In this talk, I will indicate everywhere differentiability of any viscosity solution of the above Aronsson's equation. This extends an important theorem by Evans and Smart on the infinity harmonic functions (i.e. $A$ is the identity matrix).
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In this lecture, we will explain a new method to show that regularity on the boundary of a domain implies regularity in the inside for PDE's of the Hamilton-Jacobi type. The method can be applied in different settings. One of these settings concerns continuous viscosity solutions $U : T^N\times [0,+\infty[ \rightarrow R$ of the evolutionary equation $\partial_t U(x, t) + H(x, \partial_x U(x, t) ) = 0,$ where $T^N = R^N / Z^N$, and $H: T^N \times R^N$ is a Tonelli Hamiltonian, i.e. H(x, p) is $C^2$, strictly convex superlinear in p.
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Vortices arise in many problems in condensed matter physics, including superconductivity, superfluids, and Bose-Einstein condensates. I will discuss some results on the behavior of two of these systems when there are asymptotically large numbers of vortices. The methods involve suitable renormalization of the energies both at the vortex cores and at infinity, along with a renormalization of the vortex density function.

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