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Register a new userGlobal expanding solutions of compressible Euler equations with small initial densities
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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.
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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
In this paper, we establish a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $gamma$-gas law equation of state for $gamma=2$ and the general initial density $ri in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which play a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.
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.
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.
This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evidence that one of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable.
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