Title & Abstract

Abe, Ken(Osaka Metropolitan U.)

• Title
  Existence of homogeneous Euler flows of degree −α ∉ [−2, 0]

• Abstract
  We consider (−α)-homogeneous solutions to the stationary incompressible Euler equations in R³\{0} for α ≥ 0 and in R³ for α < 0. Shvydkoy (2018) demonstrated the nonexistence of (−1)-homogeneous solutions (u, p) ∈ C¹(R³\{0}) and (−α)-homogeneous solutions in the range 0 ≤ α ≤ 2 for the Beltrami and axisymmetric flows. We consider the existence of axisymmetric (−α)-homogeneous solutions in the complementary range α ∈ R\[0, 2].

Chen, Geng(U. Kansas)

• Title
  Large solutions of compressible Euler equations

• Abstract
  Compressible Euler equations (introduced by Euler in 1757) model the motion of compressible inviscid fluids such as gases. It is well-known that solutions of compressible Euler equations often develop discontinuities, i.e. shock waves. Successful theories have been established in the past 150 years for small solutions in one space dimension. The theory on large solutions is widely open for a long time, even in one space dimension. In this talk, I will discuss some recent exciting progresses in this direction. In the first part of this talk, I will discuss our discovery of a sharp time-dependent lower bound on density. This also gives a complete resolution of shock formation problem, which extends the celebrated work of Peter Lax on small solutions in 1964. The recent progress for radially symmetric solution will be discussed. In the second part, I will show our negative example concerning the failure of current available frameworks on approximate solutions in order to establish large BV (bounded total variation) theory. The talk is based on several joint works with A. Bressan, H.K. Jenssen, R. Pan, R. Young, Q. Zhang, and S. Zhu. 

Jeong, In-Jee(Seoul National U.)

• Title
  On vorticity supported on logarithmic spirals

• Abstract
  We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on the unit circle. We show that this system is locally well-posed for L^p data as well as for atomic measures, that is logarithmic spiral vortex sheets. We prove global well-posedness for almost bounded logarithmic spirals and give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. (Joint work with Ayman Said)

Krupa, Sam(Max Planck Institute)

• Title
  Large Data Solutions to 1-D Hyperbolic Systems, Ill-Posedness, and Convex Integration

• Abstract
  For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, it is an open question if it is possible to construct solutions via convex integration. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would not have the strong traces necessary for the nonperturbative $L^2$ stability theory of Vasseur. Whether convex integration is possible is a question about large data, and the global geometric structure of genuine nonlinearity for the underlying PDE. In this talk, I will discuss recent work which shows the impossibility, for a large class of 2x2 systems, of doing convex integration via the use of $T_4$ configurations. Our work applies to every well-known 2x2 hyperbolic system of conservation laws which verifies the Liu entropy condition. This talk is based on joint work with László Székelyhidi.

Ueda, Yoshihiro(Kobe U.)

• Title
  Stability of stationary solutions for viscoelastic fluids in half space

• Abstract
  In this talk, we discuss the stability of the compressible fluid with viscoelasticity. We consider the outflow problem in a one-dimensional half-space and show the existence of a stationary solution and its stability. There exists a lot of known results for compressible fluids. In particular, the existence and stability of stationary solutions to the outflow problem were discussed in Nakamura-Nishibata-Yuge (2007) and Nakamura-Ueda-Kawashima (2010), where the Mach number was used as a criterion. Similar results are obtained for viscoelastic fluids, however, the main feature is that the criterion is constructed by the modified Mach number, which takes into account the effect of viscoelasticity. This result is based on joint research with Yusuke Ishigaki of Osaka University.

Vasseur, Alexis(U. Texas-Austin)

Lecture I

• Title 
  Inviscid limit from Navier-stokes  to small BV solutions of Euler  

• Abstract
  We show in this talk that small BV solutions to the isentropic Euler equations can be obtained as  inviscid limits from the compressible barotropic Navier-Stokes equations. This is a joint work with Geng Chen and Moon-Jin Kang. 

Lecture II

• Title Boundary vorticity estimate  for the Navier-Stokes equation and  control of layer separation in the inviscid limit 

• Abstract
  We provide a new boundary estimate on the vorticity for the incompressible Navier-Stokes equation endowed with no-slip boundary condition. The estimate is rescalable through the inviscid limit. It provides a control on the layer separation at the inviscid Kato double limit, which is consistent with the Layer separation predictions via  convex integration. This is a joint work with Jincheng Yang. 

Wang, Yi(Chinese Academy Science)

• Title
  Stability of Riemann solutions to the compressible Navier-Stokes equations

• Abstract
  The talk is concerned with our recent progress on the time-asymptotic stability of generic Riemann solutions to the 1D compressible Navier-Stokes equations (including both isentropic and full cases). Based on joint works with Moon-Jin Kang and Aleixs Vasseur.

Xu, Xiaoqian(Duke Kunshan U.)

• Title
  Mixing flow and advection-diffusion-reaction equations

• Abstract
  In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of application is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and the role it plays in the field of advection-diffusion-reaction equation, such as the famous Keller-Segel equation for chemotaxis. I will also discuss about the examples of such flows in this talk.