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Nonlinear Partial Differential Equations

Fall 2017

Date:September 19, 2017
Time:3:00pm
Location:BLOC 628
Speaker:James Kelliher, UC Riverside
Title:Nonlinear PDEs Seminar
Abstract:
Title: Bounded Vorticity, unbounded velocity solutions to the 2D Euler equations.

Abstract:
The pioneering work on bounded vorticity solutions to the 2D Euler equations was done by Yudovich in the early 1960s, working in a bounded domain. He proved existence and uniqueness of such solutions. Subsequently, existence was shown to hold in much weaker settings (vorticity lying in any Lebesgue space), but uniqueness has only ever been extended to an incrementally larger class of initial data.

Yudovich's theory extends easily to the full plane (indeed it is slightly less technical there) as long as the velocity is assumed to decay sufficiently rapidly at infinity that the Biot-Savart law holds. In 1995, Ph. Serfati established the existence and uniqueness of bounded vorticity solutions having no decay at infinity. Such solutions very much violate the Biot-Savart law, but Serfati discovered an identity that the solutions hold that can be used as a kind of substitute for that law.

The boundedness of the velocity was very important in Serfati's argument, yet there is room in his identity to accommodate some growth of the velocity at infinity. I will speak on ongoing joint work with Elaine Cozzi in which we exploit Serfati's identity to obtain existence and uniqueness classes allowing growth at infinity as large as possible (without assuming any special symmetry of the initial data). Roughly speaking, we show that existence can be achieved only for very slowly growing velocities, but that uniqueness holds for velocities growing slower than the square root of the distance from the origin. We also consider the issue of continuous dependence on initial data, which is already an interesting problem even in Yudovich's original setting.

Date:September 26, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Giles Auchmuty, University of Houston
Title:Nonlinear PDEs Seminar
Abstract:
Title: Energy Bounds for Planar Div-curl Boundary Value Problems

Abstract: This talk will describe the derivation of sharp L^2 norm estimates for the solutions of div-curl boundary value problems on bounded Lipschitz planar regions Nontrivial prescribed tangential, normal or mixed boundary condition problems are considered of the type that arise in stationary electromagnetic eld modeling. The solutions are found using special decompositions of the vector elds in terms of scalar potentials, stream functions and harmonic elds. Explicit spectral formulae for the solutions are derived that involve various eigenvalues and eigenfunctions of the Laplacian including the Steklov eigenvalues and eigenfunctions. The bounds depend on certain least eigenvalues and the given data.

Date:October 17, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Xin Liu, Texas A&M University
Title:Nonlinear PDEs Seminar
Abstract:
Title: Some gas-vacuum interface problems of compressible Navier-Stokes equations in spherically symmetric motions

Abstract:

I will talk about the well-posedness of two problems concerning the evolution of a flow connected with vacuum. The flow, or gas, connects the vacuum area in a way that the sound speed across the gas-vacuum interface has only Holder continuity. A typical example is the Lane-Emden solution for gaseous stars, where the sound speed is only 1/2-Holder continuity on the gas-vacuum interface. As pointed out by T.P. Liu in 1996, the classical hyperbolic method fails due to such singularity. Only recently, Jang and Masmoudi, Coutand, Lindblad and Shkoller independently developed some weighted energy estimates to show the well-posedness of the inviscid isentropic flows. This work is to investigate how the viscosity will help resolve such singularity. In particular, the equilibrium and the well-posedness of a model based on the thermodynamic model listed in Chandrasekhar’s book (An introduction to the study of stellar structure) is studied. Also, we investigate the global well-posedness of the Navier-Stokes equations, which allows the density and velocity to be large, the gas to connect to vacuum in a general manner but the energy to be small. This is based on my Ph.D. thesis as a student of Prof. Zhouping Xin in the Chinese University of Hong Kong, Hong Kong. ===============

Date:October 31, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Alex Mahalov, Arizona State University
Title:Nonlinear PDE's seminar
Abstract:
Title:Stochastic Three-Dimensional Navier-Stokes Equations + Waves: Averaging, Convergence, Regularity and Nonlinear Dynamics

Abstract:
We consider stochastic three-dimensional Navier-Stokes equations + Waves on long time intervals. Regularity results are established by bootstrapping from global regularity of the averaged stochastic resonant equations and convergence theorems. The averaged covariance operator couples stochastic and wave effects. The energy injected in the system by the noise is large, the initial condition has large energy, and the regularization time horizon is long. Regularization is the consequence of precise mechanisms of relevant three-dimensional nonlinear interactions. We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework. We also present theoretical and computational results for three-dimensional nonlinear dynamics.

Date:November 7, 2017
Time:3:00pm
Location:BLOC 628
Speaker:Suncica Canic, University of Houston
Title:Nonlinear PDEs Seminar
Abstract:
Title: A mathematical framework for proving existence of weak solutions to a class of fluid-structure interaction problems

Abstract:
The focus of this talk will be on nonlinear moving-boundary problems involving incompressible, viscous fluids and elastic structures. The fluid and structure are coupled via two sets of coupling conditions, which are imposed on a deformed fluid-structure interface. The main difficulty in studying this class of problems from the analysis and numerical points of view comes from the strong geometric nonlinearity due to the nonlinear fluid-structure coupling. We have recently developed a robust framework for proving existence of weak solutions to this class of problems, allowing the treatment of various structures (Koiter shell, multi- layered composite structures, mesh-supported structures), and various coupling conditions (no-slip and Navier slip). The existence proofs are constructive: they are based on the time-discretization via Lie operator splitting, and on our generalization of the famous Lions-Aubin-Simon’s compactness lemma to moving boundary problems. The constructive proof strategy can be used in the design of a loosely-coupled partitioned scheme, in which the fluid and structure sub-problems are solved separately, with the cleverly designed boundary conditions to enforce the coupling in a way that approximates well the continuous energy of the coupled problem. This provides stability and uniform energy estimates, important for the convergence proof of the numerical scheme. Applications of this strategy to the simulations of real-life problems will be shown. They include the flow of blood in a multi-layered coronary artery treated with vascular devices called stents (with Dr. Paniagua (Texas Heart Institute) and Drs. Little and Barker, Methodist Hospital, Houston), and optimal design of micro- swimmers and bio-robots (with biomed. engineer Prof. Zorlutuna, Notre Dame).

Parts of the mathematical work are joint with B. Muha

Date:November 10, 2017
Time:1:50pm
Location:BLOC 628
Speaker: Luan T. Hoang, Texas Tech University
Title: Nonlinear PDEs (joint with Mathematical Physics Seminar)
Abstract:
Title: Large-time asymptotic expansions for solutions of Navier-Stokes equations

Abstract: We study the long-time behavior of solutions to the three-dimensional Navier-Stokes equations of viscous, incompressible fluids with periodic boundary conditions. The body forces decay in time either exponentially or algebraically. We establish the asymptotic expansions of Foias-Saut-type for all Leray-Hopf weak solutions. If the force has an asymptotic expansion, as time tends to infinity, in terms of exponential functions or negative-power functions, then any weak solution admits an asymptotic expansion of the same type. Moreover, when the force's expansion holds in Gevrey spaces, which have much stronger norms than the Sobolev spaces, then so does the solution's expansion. This extends the previous results of Foias and Saut in Sobolev spaces for the case of potential forces.