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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 $a_0:=rho_0-1indot B^{frac np}_{p,1},$ $u_0indot B^{frac np-1}_{p,1}$ and $theta_0indot B^{frac np-2}_{p,1}.$ 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 $ngeq2,$ 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).
We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the
We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Leb
In this paper, we derive several new sufficient conditions of non-breakdown of strong solutions for for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that t
In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any $H^1$ initial data. The
Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $ abla_{h}{u}$ (or $ abla{u_3}$) satisfies