Weak Continuity and Compactness for Nonlinear Partial Differential Equations

TL;DR

This paper explores weak continuity and compactness in nonlinear PDEs, emphasizing compensated compactness and their applications to hyperbolic conservation laws, fluid dynamics, and differential geometry.

Contribution

It provides new insights into the weak continuity and compactness properties of solutions to nonlinear PDEs, including hyperbolic conservation laws and geometric systems.

Findings

Convergence of vanishing viscosity solutions for hyperbolic conservation laws

Analysis of weak continuity in the Gauss-Codazzi-Ricci system

Applications to the inviscid limit from Navier-Stokes to Euler equations

Abstract

We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on the compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropy flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multidimensional steady compressible fluids. We then analyze the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.