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Register a new userEnergy equalities for compressible Navier-Stokes equations
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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 hold. The method of proof is suitable for the case of periodic as well as homogeneous Dirichlet boundary conditions. In particular, by a careful analysis using the homogeneous Dirichlet boundary condition, no boundary layer assumptions are required when dealing with bounded domains with boundary.
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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.
It is well-known that a Leray-Hopf weak solution in $L^4 (0,T; L^4(Omega))$ for the incompressible Navier-Stokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lionss condition for energy balance is also valid for the weak solutions of the isentropic compressible Navier-Stokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the non-homogenous incompressible Navier-Stokes equations. This is an improvement of corresponding results obtained by Yu in [32, Arch. Ration. Mech. Anal., 225 (2017)], and our criterion via the gradient of the velocity partially answers a question posed by Liang in [18, Proc. Roy. Soc. Edinburgh Sect. A (2020)].
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 equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by $epsilon$ tends to 0.
We prove the existence of relative finite-energy vanishing viscosity solutions of the one-dimensional, isentropic Euler equations under the assumption of an asymptotically isothermal pressure law, that is, $p(rho)/rho = O(1)$ in the limit $rho to infty$. This solution is obtained as the vanishing viscosity limit of classical solutions of the one-dimensional, isentropic, compressible Navier--Stokes equations. Our approach relies on the method of compensated compactness to pass to the limit rigorously in the nonlinear terms. Key to our strategy is the derivation of hyperbolic representation formulas for the entropy kernel and related quantities; among others, a special entropy pair used to obtain higher uniform integrability estimates on the approximate solutions. Intricate bounding procedures relying on these representation formulas then yield the required compactness of the entropy dissipation measures. In turn, we prove that the Young measure generated by the classical solutions of the Navier--Stokes equations reduces to a Dirac mass, from which we deduce the required convergence to a solution of the Euler equations.
In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier-Stokes equations. Enlightening regular criteria via pressure $Pi=frac{text {divdiv}}{-Delta}(u_{i}u_{j})$ of the 3D incompressible Navier-Stokes equations on bounded domain, we generalize Beirao da Veigas result in [1] from the incompressible Navier-Stokes equations to the isentropic compressible Navier-Stokes system in the case away from vacuum.
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