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Register a new userExpanding large global solutions of the equations of compressible fluid mechanics
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In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the three-dimensional isentropic compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent $gamma$ belongs to the interval $(1,frac53]$, then these affine motions are nonlinearly stable. Small perturbations lead to globally-in-time defined solutions that remain in the vicinity of the manifold of affine motions, they remain smooth in the interior of their support, and no shocks are formed in the process. Our strategy relies on two key ingredients. We first provide a new interpretation of the affine motions using an (almost) invariant action of GL$^+(3)$ on the compressible Euler system. This transformation dictates a particular rescaling of time and a change of variables, which in turn exposes a stabilization mechanism induced by the expansion of the background affine motions when $gammain(1,frac53]$. We then switch to a Lagrangian description of the rescaled Euler system which reflects the geometry of expanding affine motions. We introduce new ideas with respect to the existing well-posedness frameworks and build high-order energy spaces to prove global-in-time stability, thereby making crucial use of the new stabilization effect
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We prove the existence of a large class of global-in-time expanding solutions to vacuum free boundary compressible Euler flows without relying on the existence of an underlying finite-dimensional family of special affine solutions of the flow.
In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key element of our proof is the control of a particular defect measure associated to the pressure which avoids the use of the eective ux. Using this new tool, we solve an open problem namely global existence of solutions {`a} la Leray for such a system without assuming any restriction on the anisotropy amplitude. It provides a exible and natural way to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.
In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.
In this note, we prove that the solutions obtained to the spherically symmetric Euler equations in the recent works [2, 3] are weak solutions of the multi-dimensional compressible Euler equations. This follows from new uniform estimates made on the artificial viscosity approximations up to the origin, removing previous restrictions on the admissible test functions and ruling out formation of an artificial boundary layer at the origin. The uniform estimates may be of independent interest as concerns the possible rate of blow-up of the density and velocity at the origin for spherically symmetric flows.
In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping $$ partial_trho+operatorname{div}(rho u)=0, quad partial_t(rho u)+operatorname{div}left(rho uotimes u+p,I_{3}right)=-,frac{mu}{(1+t)^{lambda}},rho u, quad rho(0,x)=bar rho+varepsilonrho_0(x),quad u(0,x)=varepsilon u_0(x), $$ where $xinmathbb R^3$, $mu>0$, $lambdageq 0$, and $barrho>0$ are constants, $rho_0,, u_0in C_0^{infty}(mathbb R^3)$, $(rho_0, u_0) otequiv 0$, $rho(0,cdot)>0$, and $varepsilon>0$ is sufficiently small. For $0leqlambdaleq1$, we show that there exists a global smooth solution $(rho, u)$ when $operatorname{curl} u_0equiv 0$, while for $lambda>1$, in general, the solution $(rho, u)$ will blow up in finite time. Therefore, $lambda=1$ appears to be the critical value for the global existence of small amplitude smooth solutions.
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