On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces

We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in $\mathbb{R}^n.$ We focus on the so-called critical Besov regularity framework. In this setting, it is natural to consider initial densities $\rho_0,$ velocity fields $u_0$ and temperatures $\theta_0$ with $a_0:=\rho_0-1\in\dot B^{\frac np}_{p,1},$ $u_0\in\dot B^{\frac np-1}_{p,1}$ and $\theta_0\in\dot B^{\frac np-2}_{p,1}.$ After recasting the whole system in Lagrangian coordinates, and working with the \emph{total energy along the flow} rather than with the temperature, we discover that the system may be solved by means of Banach fixed point theorem in a critical functional framework whenever the space dimension is $n\geq2,$ and $1<p<2n.$ Back to Eulerian coordinates, this allows to improve the range of $p$'s for which the system is locally well-posed, compared to Danchin, Comm. Partial Differential Equations 26 (2001)

圧縮性Navier-Stokes 方程式と関連する問題の臨界空間における適切性

要約のみTohoku University小川卓克課

A note on the well-posedness of the compressible viscous fluid in the critical Besov space (Regularity and Singularity for Partial Differential Equations with Conservation Laws)

"Regularity and Singularity for Partial Differential Equations with Conservation Laws". June 3～5, 2015. edited by Keiichi Kato, Mishio Kawashita, Masashi Misawa and Takayoshi Ogawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We prove well-posedness of the compressible Navier-Stokes system in the Lagrangian formulation by the use of absolute temperature. The purpose of this article is to illustrate the difference between the use of the total energy along the flow in Chikami-Danchin [3] (J. Difff. Eq, 258 (2015), 3435-3467) and the absolute temperature

Some a priori estimate related to the well-posedness for the barotropic compressible Navier-Stokes system (Harmonic Analysis and Nonlinear Partial Differential Equations)

"Harmonic Analysis and Nonlinear Partial Differential Equations". July 4～6, 2016. edited by Hideo Kubo and Hideo Takaoka. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider some global a priori estimate related to the well-posedness for the barotropic compressible Navier-Stokes system. We employ the Fourier-Herz spaces instead of the standard Besov spaces, and show a global in time a priori bound of the solutions for the initial data in the Lq-type regularity framework

A note on the decay estimates for the compressible Navier-Stokes-Poisson system in critical Besov spaces (Mathematical Analysis of Viscous Incompressible Fluid)

This is a survey on the Cauchy problem for the Navier-Stokes-Poisson system in the critical regularity framework. Under a suitable additional condition involving only the low frequencies of the data, we establish optimal decay estimates in the L^{2}-critical framework for the global solutions around small perturbations of a linearly stable constant state

On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system

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