On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping

In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping \begin{equation*} \partial_t\rho+\operatorname{div}(\rho u)=0, \quad \partial_t(\rho u)+\operatorname{div}\left(\rho u\otimes u+p\,I_d\right)=-\alpha(t)\rho u, \quad \rho(0,x)=\bar \rho+\varepsilon\rho_0(x),\quad u(0,x)=\varepsilon u_0(x), \end{equation*} where $x=(x_1, \cdots, x_d)\in\Bbb R^d$ $(d=2,3)$, the frictional coefficient is $\alpha(t)=\frac{\mu}{(1+t)^\lambda}$ with $\lambda\ge0$ and $\mu>0$, $\bar\rho>0$ is a constant, $\rho_0,u_0 \in C_0^\infty(\Bbb R^d)$, $(\rho_0,u_0)\not\equiv 0$, $\rho(0,x)>0$, and $\varepsilon>0$ is sufficiently small. One can totally divide the range of $\lambda\ge0$ and $\mu>0$ into the following four cases: Case 1: $0\le\lambda0$ for $d=2,3$; Case 2: $\lambda=1$, $\mu>3-d$ for $d=2,3$; Case 3: $\lambda=1$, $\mu\le 3-d$ for $d=2$; Case 4: $\lambda>1$, $\mu>0$ for $d=2,3$. \noindent We show that there exists a global $C^{\infty}-$smooth solution $(\rho, u)$ in Case 1, and Case 2 with $\operatorname{curl} u_0\equiv 0$, while in Case 3 and Case 4, in general, the solution $(\rho, u)$ blows up in finite time. Therefore, $\lambda=1$ and $\mu=3-d$ appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution $(\rho, u)$ in $d-$dimensional compressible Euler equations with time-depending damping.Comment: 32 pages, 2 figure

On global multidimensional supersonic flows with vacuum states at infinity

In this paper, we are concerned with the global existence and stability of a smooth supersonic flow with vacuum state at infinity in a 3-D infinitely long divergent nozzle. The flow is described by a 3-D steady potential equation, which is multi-dimensional quasilinear hyperbolic (but degenerate at infinity) with respect to the supersonic direction, and whose linearized part admits the form \p_t^2-\ds\f{1}{(1+t)^{2(\g-1)}}(\p_1^2+\p_2^2)+\ds\f{2(\g-1)}{1+t}\p_t for 1<\g<2. From the physical point of view, due to the expansive geometric property of the divergent nozzle and the mass conservation of gas, the moving gas in the nozzle will gradually become rarefactive and tends to a vacuum state at infinity, which implies that such a smooth supersonic flow should be globally stable for small perturbations since there are no strong resulting compressions in the motion of the flow. We will confirm such a global stability phenomena by rigorous mathematical proofs and further show that there do not exist vacuum domains in any finite part of the nozzle.Comment: 46 page

The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, III: the 3-D Boltzmann equation

This paper is a continuation of the works in \cite{Euler} and \cite{NS}, where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In the present paper, we will confirm this physical phenomenon for the Boltzmann equation by obtaining the exact lower and upper bound on the macroscopic density function.Comment: 41 page

The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I: 3D Euler equations

We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. In this paper, as the first part of our three papers, we will confirm this physical phenomenon for the compressible inviscid fluids by obtaining the exact lower and upper bound on the density function.Comment: 55 page

On the global existence and stability of 3-D viscous cylindrical circulatory flows

In this paper, we are concerned with the global existence and stability of a 3-D perturbed viscous circulatory flow around an infinite long cylinder. This flow is described by 3-D compressible Navier-Stokes equations. By introducing some suitably weighted energy spaces and establishing a priori estimates, we show that the 3-D cylindrical symmetric circulatory flow is globally stable in time when the corresponding initial states are perturbed suitably small.Comment: 21 page

On the lifespan of and the blowup mechanism for smooth solutions to a class of 2-D nonlinear wave equations with small initial data

This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation \p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu) $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and $(c_1'(0))^2+(c_2'(0))^2\neq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition (u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x)) with $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$, and \ve>0 is small, we will show that the classical solution $u(t,x)$ stops to be smooth at some finite time T_{\ve}. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives \na_{t,x}u(t,x), while $u(t,x)$ itself is continuous up to the blowup time T_{\ve}.Comment: 22 page

On the blowup and lifespan of smooth solutions to a class of 2-D nonlinear wave equations with small initial data

We are concerned with a class of two-dimensional nonlinear wave equations \p_t^2u-\div(c^2(u)\na u)=0 or \p_t^2u-c(u)\div(c(u)\na u)=0 with small initial data (u(0,x),\p_tu(0,x))=(\ve u_0(x), \ve u_1(x)), where $c(u)$ is a smooth function, $c(0)\not =0$, $x\in\Bbb R^2$, $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$ depend only on $r=\sqrt{x_1^2+x_2^2}$, and \ve>0 is sufficiently small. Such equations arise in a pressure-gradient model of fluid dynamics, also in a liquid crystal model or other variational wave equations. When $c'(0)\not= 0$ or $c'(0)=0$, $c"(0)\not= 0$, we establish blowup and determine the lifespan of smooth solutions.Comment: 30 pages, 2 figure

Blowup of smooth solutions for general 2-D quasilinear wave equations with small initial data

For the 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(\nabla u)\partial_{ij}u=0$ with coefficients independent of the solution $u$, a blowup result for small data solutions has been established in [1,2] provided that the null condition does not hold and a generic nondegeneracy condition on the initial data is fulfilled. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(u, \nabla u)\partial_{ij}u=0$ with coefficients that depend simultaneously on $u$ and $\nabla u$. When the null condition does not hold and a suitable nondegeneracy condition on the initial data is satisfied, we show that smooth small data solutions blow up in finite time. Furthermore, we derive an explicit expression for the lifespan and establish the blowup mechanism.Comment: 27 page

Large time asymptotic behavior of the compressible Navier-Stokes Equations in partial Space-Periodic Domains

In this paper, we study the large time behavior of the 3-D isentropic compressible Navier-Stokes equation in the partial space-periodic domains, and simultaneously show that the related profile systems can be described by like Navier-Stokes equations with suitable "pressure" functions in lower dimensions. Our proofs are based on the energy methods together with some delicate analysis on the corresponding linearized problems.Comment: 28 pag

Global multidimensional shock waves for 2-D and 3-D unsteady potential flow equations

Although local existence of multidimensional shock waves has been established in some fundamental references, there are few results on the global existence of those waves except the ones for the unsteady potential flow equations in n-dimensional spaces (n > 4) or in special unbounded space-time domains with non-physical boundary conditions. In this paper, we are concerned with both the local and global multidimensional conic shock wave problem for the unsteady potential flow equations when a pointed piston (i.e., the piston degenerates into a single point at the initial time) or an explosive wave expands fast in 2-D or 3-D static polytropic gas. It is shown that a multidimensional shock wave solution of such a class of quasilinear hyperbolic problems not only exists locally, but it also exists globally in the whole space-time and approaches a self-similar solution as t goes to infinity.Comment: 70 pages, 3 figure