Existence of global strong solution for Korteweg system with large infinite energy initial data

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:438

Article

Haspot, Boris¹ 

[1] Univ Paris 09, PSL Res Univ, CEREMADE, CNRS,Umr 7534, Pl Marechal Lattre Tassigny, F-75775 Paris 16, France

关键词: Korteweg system;    Strong solution;    Besov spaces;    Critical spaces;    Paraproduct;   

DOI  :  10.1016/j.jmaa.2016.01.047

来源: Elsevier

【 摘 要 】

This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E. Dunn and J. Serrin (1985) (see [18]), which can be used as a phase transition model. We aim at proving the existence of local and global (under a condition of smallness on the initial data) strong solutions with initial density ln rho(0), belonging to the Besov space B-2,infinity(N/2). It implies in particular that some classes of discontinuous initial density generate strong solutions. The proof relies on the fact that the density can be written as the sum of the solution rho(L), of the associated linear system and a remainder term (rho) over bar; this last term is more regular than rho(L) provided that we have regularizing effects induced on the bilinear convection term. The main difficulty consists in obtaining new estimates of maximum principle type for the associated linear system; this is based on a characterization of the Besov space in terms of the semi-group associated with this linear system. We show in particular the existence of global strong solution for small initial data in ((B) over tilde (N/2)(2,infinity)-1 boolean AND L-infinity) x B-2,infinity(N/2)-1; it allows us to exhibit a family of large energy initial data when N = 2 providing global strong solution. In conclusion we introduce the notion of quasi-solutions for the Korteweg's system (a tool which has been developed in the framework of the compressible Navier-Stokes equations [31,30,32,26,27]) which enables to obtain the existence of global strong solution with a smallness condition which is subcritical. Indeed we can deal with large initial velocity in B-2,1(N/2)-1. As a corollary, we get global strong solution for highly compressible Korteweg system when N >= 2. It means that for any large initial data (under an irrotational condition on the initial velocity) we have the existence of global strong solution provided that the Mach number is sufficiently large. (C) 2016 Elsevier Inc. All rights reserved.

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