Day 1 (June 10)

 In this talk, we will discuss relativistic quantum hydrodynamics equations from the Klein-Gordon equation with Poisson effects via the Madelung transformation, and establish the local existence of classical solutions with Cauchy data. This framework provides new perspectives on singular limits, including the semi-classical and non-relativistic framework. Future work will be devoted to rigorous analysis of these limiting processes.

 Wave turbulence refers to the statistical theory of weakly nonlinear dispersive waves. In the weakly turbulent regime of a system of dispersive waves, its statistics can be described via a coarse-grained dynamics, governed by the kinetic wave equation. In this talk, we will briefly introduce wave turbulence theory, and see some applications. The talk is based on a joint work with Pierre Germain (ICL) and Katherine Zhiyuan Zhang (Northeastern), and a joint work with Pierre Germain and Angeliki Menegaki (ICL). 

 When the steady flows are away from stagnation, the associated Euler equations can be locally reduced to a semilinear equation. On the other hand, stagnation of flows is not only an interesting phenomenon in fluid mechanics, but also plays a significant role in understanding many important properties of fluid equations. It also induces many challenging problems in analysis. First, we give a classification of incompressible Euler flows via the set of flow angles. Second, we discuss the senario when the Euler equations can be reduced to a single semilinear equation in terms of stream function. The applications for these classifications will also be addressed. 

 The structure stability of stationary transonic shocks plays a crucial role in the aerodynamic design of supersonic nozzles. In this talk, I will report that mass-additions stabilize transonic shocks in steady compressible Euler flows within two dimensional straight nozzles. The presence of a source term in the balance of mass prevents the widely used method of Lagrange transform or stream functions. Moreover, massaddion leads to background solutions of the system being nontrivial large functions. To overcome these difficulties, we propose a novel decomposition for the subsonic Euler equations, that effectively separates the main components of the elliptic and hyperbolic modes. The new approach necessitates solving a Dirichlet-Neumann-Venttselmixed boundary value problem of a second order elliptic equation of pressure containing various integral and pointwise nonlocal terms, and transport equations of entropy and total enthalpy, as well as a family of two point boundary value problems of ordinary differential equations on tangential velocity in each cross section of the nozzle. This approach allows us to clarify the mathematical mechanisms that promote shock stability. We establish the structure stability of almost every background solution under small perturbations of the inlet supersonic flows and outlet back pressures. All the physical quantities possess the same regularity. It is expected that the approach may be applicable to treat other free boundary problems associated with nonlinear systems of conservation laws of elliptic hyperbolic composite mixed type. This is a joint work with Dr. Junlei Gao.

 We consider the compressible Navier-Stokes-Fourier (NSF) equations on a half line and investigate the time asymptotic behavior toward the outgoing viscous shock wave. Specifically, we address boundary problems in the outflow setting, where the velocity and temperature at the boundary are held constant. When the asymptotic profile, determined by the prescribed constant states at both the boundary and far field, forms a viscous shock, we demonstrate that the solution converges uniformly in space to the shifted viscous shock profile, assuming the initial perturbation is sufficiently small in the H^1 norm. Notably, we do not require a zero-mass condition on the initial data. Our result is derived using the a-contraction method with shifts and it is the first result to establish the stability of the viscous shock for the outflow problem for NSF equations on the half-space.

 We discuss the unique existence of an axisymmetric supersonic solution with nonzero vorticity and nonzero angular momentum density for the steady Euler-Poisson system in three-dimensional divergent nozzles when prescribing the velocity, strength of electric field, and the entropy at the entrance. We first reformulate the problem via the method of the Helmholtz decomposition for three-dimensional axisymmetric flows and obtain a solution to the reformulated problem by the iteration method. We deal carefully with singularity issues related to the polar angle on the axis of the divergent nozzle.

 For a uniformly supersonic flow past a convex cornered wedge with the pressure being given for the surrounding quiescent gas at the downstream, as shown in experiemental results, it is expected to form a shock followed with a contact discontinuity, which is also called the jet flow. By the shock polar analysis, it is well-known that there are two possible shocks, one is strong shock and the other one is the weak shock. The strong shock is always transonic, while the weak shock could be transonic or supersonic. In this paper, we prove the global existence, asymptotic behaviors, uniqueness and stability of the subsonic jet with a strong transonic shock under the perturbation of the upstream flow and the pressure of the surrounding quiescent gas, for the two-dimensional stready full Euler equations. We first formulate the problem into a nonlinear problem with two free boundaries meeting at the wedge corner, and formulate the boundary conditions on them. Then we introduce a modified Lagrange coordinates transformation to straighten the two free boundaries at the same time, and study the elliptic estimate with proper weighted Holer norms to deal with the wedge corner singularity and the asymptotic behaviours for the Euler equations in the Lagrange coordinates carefully, and then design an iteration scheme based on the estimates.