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Series: PDE Seminar

Series: PDE Seminar

Geometric tangential analysis refers to a constructive systematic approach based on the concept that a problem which enjoys greater regularity can be “tangentially" accessed by certain classes of PDEs. By means of iterative arguments, the method then imports regularity, properly corrected through the path used to access the tangential equation, to the original class. The roots of this idea likely go back to the foundation of De Giorgi’s geometric measure theory of minimal surfaces, and accordingly, it is present in the development of the contemporary theory of free boundary problems. This set of ideas also plays a decisive role in Caffarelli’s work on fully non-linear elliptic PDEs, and subsequently in his studies on Monge-Ampere equations from the 1990’s. In recent years, however, geometric tangential methods have been significantly enhanced, amplifying their range of applications and providing a more user-friendly platform for advancing these endeavors. In this talk, I will discuss some fundamental ideas supporting (modern) geometric tangential methods and will exemplify their power through select examples.

Series: PDE Seminar

The aim of talk is threefold. First, we solve the cubic nonlinear Schr\"odinger equation on the real line with initial data a sum of Dirac deltas. Secondly, we show a Talbot effect for the same equation. Finally, we prove an intermittency phenomena for a class of singular solutions of the binormal flow, that is used as a model for the vortex filaments dynamics in 3-D fluids and superfluids. If time permits some questions concerning the transfer of energy and momentum will be also considered.

Series: PDE Seminar

Almost all biological activities involve transport and distribution of ions and charged particles. The complicated coupling and competition between different ionic solutions in various biological environments give the intricate specificity and selectivity in these systems. In this talk, I will introduce several extended general diffusion systems motivated by the study of ion channels and ionic solutions in biological cells. In particular, I will focus on the interactions between different species, the boundary effects and in many cases, the thermal effects.

Series: PDE Seminar

The magnetohydrodynamic (MHD) equations govern the motion of electrically conducting fluids such as plasmas, liquid metals, and electrolytes. They consist of a coupled system of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. Besides their wide physical applicability, the MHD equations are also of great interest in mathematics. They share many similar features with the Navier-Stokes and the Euler equations. In the last few years there have been substantial developments on the global regularity problem concerning the magnetohydrodynamic (MHD) equations, especially when there is only partial or fractional dissipation. The talk presents recent results on the global well-posedness problem for the MHD equations with various partial or fractional dissipation.

Series: PDE Seminar

I will review recent results on small scale creation in solutions of the Euler equation. A numerical simulation due to Hou and Luo suggests a new scenario for finite time blow up in three dimensions. A similar geometry in two dimensions leads to examples with very fast, double exponential in time growth in the gradient of vorticity. Such growth is know to be sharp due to upper bounds going back to 1930s. If I have time, I will also discuss several models that have been proposed to help understand the three-dimensional case.

Series: PDE Seminar

The talk is about a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain. This is a joint work with R. Gong and A. Swiech.

Series: PDE Seminar

The Cucker-Smale system is a popular model of collective behavior of interacting agents, used, in particular, to model bird flocking and fish swarming. The underlying premise is the tendency for a local alignment of the bird (or fish, or ...) velocities. The Euler-Cucker-Smale system is an effective macroscopic PDE limit of such particle systems. It has the form of the pressureless Euler equations with a non-linear density-dependent alignment term. The alignment term is a non-linear version of the fractional Laplacian to a power alpha in (0,1). It is known that the corresponding Burgers' equation with a linear dissipation of this type develops shocks in a finite time. We show that nonlinearity enhances the dissipation, and the solutions stay globally regular for all alpha in (0,1): the dynamics is regularized due to the nonlinear nature of the alignment. This is a joint work with T. Do, A.Kiselev and C. Tan.

Series: PDE Seminar

In this talk, a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.

Series: PDE Seminar

In 1944, L.D. Landau first discovered explicit (-1)-homogeneous solutions of 3-d stationary incompressible Navier-Stokes equations (NSE) with precisely one singularity at the origin, which are axisymmetric with no swirl. These solutions are now called Landau solutions. In 1998 G. Tian and Z. Xin proved that all solutions which are (-1) homogeneous, axisymmetric with one singularity are Landau solutions. In 2006 V. Sverak proved that with just the (-1)-homogeneous assumption Landau solutions are the only solutions with one singularity. He also proved that there are no such solutions in dimension greater than 3. Our work focuses on the (-1)-homogeneous solutions of 3-d incompressible stationary NSE with finitely many singularities on the unit sphere.In this talk we will first classify all (-1)-homogeneous axisymmetric no-swirl solutions of 3-d stationary incompressible NSE with one singularity at the south pole on the unit sphere as a two dimensional solution surface. We will then present our results on the existence of a one parameter family of (-1)-homogeneous axisymmetric solutions with non-zero swirl and smooth on the unit sphere away from the south pole, emanating from the two dimensional surface of axisymmetric no-swirl solutions. We will also present asymptotic behavior of general (-1)-homogeneous axisymmetric solutions in a cone containing the south pole with a singularity at the south pole on the unit sphere. We also constructed families of solutions smooth on the unit sphere away from the north and south poles.This is a joint work with Professor Yanyan Li and Li Li.