#### ANALYSIS AND PDE SEMINAR

##### Fridays, 2:00-3:00pm, KAP414

For questions, contact one of the organizers: Juhi Jang, Igor Kukavica, Nabil Ziane, or Christian Zillinger

# Thursday, January 12, 2017, 3:30-4:30pm in KAP 245

## Polona Durcik, University of Bonn

### Entangled multilinear forms and applications

Abstract: We discuss L^p estimates for some multilinear singular integral forms and their applications to sharp quantitative norm convergence of ergodic averages with respect to two commuting transformations, quantitative cancellation estimates for the simplex Hilbert transform, and a question on side lengths of corners in dense subsets of the Euclidean space.

# Friday, February 10, 2017

## Eun Heui Kim, California State University, Long Beach

### Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions

Abstract: We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for the two dimensional Riemann problems. We establish the existence result of the transonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock.

# Friday, March 10, 2017

## Ming Chen, University of Pittsburgh

### Solitary stratified water waves

Abstract: We consider 2D steady water waves with heterogeneous density. The presence of stratification allows for a wide variety of traveling waves, including fronts, so-called generalized solitary waves with ripples in the far field, and even fronts with ripples! Among these many possible wave patterns, we prove that for any smooth choice of upstream velocity and monotone streamline density function, there always exists a continuous curve of solitary waves with large amplitude, which are even and decreasing monotonically on either side of a central crest. As one moves along this curve, the horizontal fluid velocity comes arbitrarily close to the wave speed. We will also discuss a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the Froude number from above and below that are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores for stratified surface waves; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry. This is a joint work with Sam Walsh and Miles Wheeler.

# Friday, March 24, 2017

## Alpar Richard Meszaros, UCLA

### Density constraints and congestion effects in advection-diffusion equations

Abstract: In this talk I will present how the theory of optimal transport can be used to study some PDE systems (not necessarily having a gradient flow structure) describing the evolution of the density of a population subject to a density constraint. In this context a particular emphasis will be on a Fokker-Planck type equation. From the modeling point of view this equation can describe the movement of a crowd in a bounded domain, when individuals try to follow a given spontaneous velocity field, but are subject to a Brownian diffusion and - to avoid congestion - have to adapt to a density constraint. From the mathematical point of view, a pressure gradient appears in the PDE (active only in the saturated zones) which affects the movement. The presented results are based on joint works with F. Santambrogio (Paris-Sud, Orsay) and with S. Di Marino (SNS, Pisa).

# Friday, March 31, 2017

## Marcelo Disconzi, Vanderbilt University

### The three-dimensional free boundary Euler equations with large surface tension.

Abstract: We study the incompressible free boundary Euler equations with surface tension in three spatial dimensions. We establish existence and uniqueness of solutions to the initial value problem. Then, we prove that under natural assumptions, when the coefficient of surface tension tends to infinity, solutions to the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary. Finally, we present new a priori estimates for smooth solutions to the free-boundary equations

# Friday, April 14, 2017, 4:30pm

## Alexey Miroshnikov, UCLA

### title: On the problem of dynamic cavitation in nonlinear elasticity

Abstract: In this work we study the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. Cavitating solutions were introduced by J.M. Ball (1982) in elastostatics and by K.A. Pericak-Spector and S. Spector (1988) in elastodynamics. They turn out to decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (in the sense of hyperbolic conservation laws) for polyconvex energies. In our work we established various further properties of cavitating solutions. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions d = 2,3 we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress-free cavities the critical stretching associated with dynamically cavitating solutions coincides with the critical stretching in the bifurcation diagram of equilibrium elasticity. This is a joint work with A. Tzavaras (KAUST).

# Friday, April 21, 2017

## Allen Yilun Wu, Brown University

### title: Rotating star solutions to the Euler-Poisson and Vlasov-Poisson equations

Abstract: The equilibrium shape and density distribution of rotating fluid under self-gravitation is a classical problem in mathematical physics. Early efforts beginning in the eighteenth century were devoted to finding ellipsoidal shapes with constant density. In the twentieth century, major progress was made by studying steady rotating solutions to the compressible Euler-Poisson equations. Two methods of constructing solutions have been used. Assuming a polytropic equation of state $p=\rho^\gamma$, a variational method, pioneered by the work of Auchmuty and Beals, proves existence of solutions if $\gamma>\frac43$. On the other hand, we present a perturbative result that establishes existence for $\gamma>\frac65$. The method is built upon an old work of Lichtenstein. We also prove an analogous result for the Vlasov-Poisson equations modeling a similar physical problem.

# Friday, October 7, 2016

## Christian Zillinger, USC

### An Introduction to Phase-mixing

Abstract: In this talk I will give a brief introduction to the phase-mixing mechanism and how it leads to damping. Here, a focus will be on linear Landau damping and the role of regularity. If time permits, I will also discuss linear inviscid damping for the 2D Euler equations.

# Friday, October 14, 2016

## Gang Zhou, Caltech

### Exponential Convergence to the Maxwell Distribution of Solutions of Spatially Inhomogenous Boltzmann Equations

Abstract: In this talk I will present a recent proof of a conjecture of C. Villani, namely the exponential convergence of solutions of spatially inhomogenous Boltzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians.

# Friday, November 4, 2016

## Juhi Jang, USC

### Expanding large global solutions of compressible Euler equations

Abstract: I will present a necessary condition for global-in-time existence of smooth compressible inviscid gas/fluid flows having the finite total mass and the finite total energy and give an example of the solutions to Euler equations satisfying such a condition, which were recently constructed by Sideris. Such solutions are compactly supposed, satisfy the physical vacuum condition, and expand into the vacuum. Global-in-time stability of such solutions will be discussed. This is based on joint work with Mahir Hadzic.

# Friday, November 11, 2016

## Yifeng Yu, UC Irvine

### G-equation in the modeling of flame propagation

Abstract: G-equation is a well known model in turbulent combustion. In this talk, I will present some works about how the effective burning velocity (turbulent flame speed) depends on the strength of the ambient fluid (e.g. the speed of the wind) under various G-equation model.

# Friday, November 18, 2016

## Fei Wang, USC

### Sobolev stability threshold for 2D shear flows near Couette

Abstract: We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\| U(y) - y\|_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon \ll \nu^{1/2}$, where $\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-close in $H^1$ to $(e^{t \nu \partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \gg \nu^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than $\nu^{1/2}$ for 2D shear flows close to the Couette flow. This is a joint work with Jacob Bedrossian and Vlad Vicol.

# Friday, December 5, 2016

## Boris Muha, University of Zagreb

### Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition

Abstract: We study a nonlinear, moving boundary fluid-structure interaction (FSI) problem between an incompressible, viscous Newtonian fluid, modeled by the 2D Navier-Stokes equations, and an elastic structure modeled by the shell or plate equations. The fluid and structure are coupled via the Navier slip boundary condition and balance of contact forces at the fluid- structure interface. The slip boundary condition is more realistic than the classical no-slip boundary condition in situations, e.g., when the structure is “rough”, and in modeling FSI dynamics near, or at a contact. Cardio-vascular tissue and cell-seeded tissue constructs, which consist of grooves in tissue scaffolds that are lined with cells, are examples of “rough” elastic interfaces interacting with an incompressible, viscous fluid. The problem of heart valve closure is an example of a FSI problem with a contact involving elastic interfaces. We prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting. This is a joint work with S.~Canic, University of Houston.