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| Weak Continuity and Compactness for Nonlinear Partial DifferentialEquations |
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Citation： |
Gui-Qiang G.CHEN.Weak Continuity and Compactness for Nonlinear Partial DifferentialEquations[J].Chinese Annals of Mathematics B,2015,36(5):715~736 |
| Page view： 1136
Net amount： 1083 |
Authors： |
Gui-Qiang G.CHEN; |
Foundation： |
This work was supported by the UK EPSRC Science and
Innovation Award to the Oxford Centre for Nonlinear PDE
(No.,EP/E035027/1), the UK EPSRC Award to the EPSRC Centre for
Doctoral Training in PDEs (No.,EP/L015811/1), the National Natural
Science Foundation of China (No.,10728101) and the Royal
Society-Wolfson Research Merit Award (UK). |
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| Abstract： |
This paper presents 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. The compactness and
convergence of vanishing viscosity solutions for nonlinear
hyperbolic conservation laws are first analyzed, including the
inviscid limit from the Navier-Stokes equations to the Euler
equations for homentropic 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
multi-dimensional steady compressible fluids. Then the weak
continuity and rigidity of the Gauss-Codazzi-Ricci system and
corresponding isometric embeddings in differential geometry are
revealed. Further references are also provided for some recent
developments on the weak continuity and compactness for nonlinear
partial differential equations. |
Keywords： |
Weak continuity, Compensated compactness, Nonlinear partial
differential equations, Euler equations, Gauss-Codazzi-Ricci system |
Classification： |
35-02, 35A35, 35A01, 35L65, 35M10, 35M30, 35D30, 53C24, 53C42 |
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