49 research outputs found
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 , the
frictional coefficient is with
and , is a constant, , , , and
is sufficiently small. One can totally divide the range of
and into the following four cases:
Case 1: for ;
Case 2: , for ;
Case 3: , for ;
Case 4: , for .
\noindent We show that there exists a global smooth solution
in Case 1, and Case 2 with , while
in Case 3 and Case 4, in general, the solution blows up in finite
time. Therefore, and appear to be the critical power and
critical value, respectively, for the global existence of small amplitude
smooth solution in 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) , where , , and .
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 , and \ve>0 is small, we will show that the classical solution
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 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 is a
smooth function, , , depend only on , 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 or , , 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 with coefficients independent
of the solution , 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
with
coefficients that depend simultaneously on and . 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