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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.
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.
We consider the isothermal Euler system with damping. We rigorously show the convergence of Barenblatt solutions towards a limit Gaussian profile in the isothermal limit $gamma$ $rightarrow$ 1, and we explicitly compute the propagation and the behavi
Global existence for the nonisentropic compressible Euler equations with vacuum boundary for all adiabatic constants $gamma > 1$ is shown through perturbations around a rich class of background nonisentropic affine motions. The notable feature of the
In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Fu
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 a