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Weak continuity and compactness for nonlinear partial differential equations

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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.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Gui-Qiang G. Chen

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Gui-Qiang G. Chen

  3. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Gui-Qiang G. Chen

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In Honor of the Scientific Contributions of Professor Luc Tartar

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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Chen, GQ.G. Weak continuity and compactness for nonlinear partial differential equations. Chin. Ann. Math. Ser. B 36, 715–736 (2015). https://doi.org/10.1007/s11401-015-0973-x

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