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In this paper, we construct global weak solutions{`a} la Hoff (i.e. intermediate regularity) for the compressible Navier-Stokes system governing a barotropic fluid with a pressure law p($rho$) = a$rho$ $gamma$ where a > 0 and $gamma$ $ge$ d/(4 -- d)) and with an anisotropic fourth order symmetric viscous tensor with smooth coefficients under the assumption that the norms of the initial data ($rho$0 -- M, u0) $in$ L 2$gamma$ T d x (H 1 (T d)) d are sufficiently small, where M denotes the total mass of the fluid. We consider periodic boundary conditions for simplicity i.e. a periodic box $Omega$ = T d with d = 2, 3 with |$Omega$| = 1. The main technical contribution of our paper is the extension of the Hoff solutions framework by relaxing the integrability needed for the initial density which is usually assumed to be L $infty$ (T d). In this way, we are able to cover the case of viscous tensors that depend on the time and space variables. Moreover, when comparing to the results known for the global weak solutions{`a} la Leray (i.e. obtained assuming only the basic energy bounds), we obtain a relaxed condition on the range of admissible adiabatic coefficients $gamma$.
We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.
In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal
The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these equalities to hol
In this paper we consider the barotropic compressible quantum Navier-Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanish. Following recent works on degenerate compressible Navier-Stokes equa
In this paper we give optimal lower bounds for the blow-up rate of the $dot{H}^{s}left(mathbb{T}^3right)$-norm, $frac{1}{2}<s<frac{5}{2}$, of a putative singular solution of the Navier-Stokes equations, and we also present an elementary proof for a l