• Abstract

The paper investigates the well-posedness of global solutions and the existence of global attractors for weakly damped FDS nonlinear wave equations. It establishes the well-posedness of weak solutions using Galerkin approximation and a priori estimate. Subsequently, a dynamical system is constructed based on the well-posedness of the solution. The existence of a bounded absorbing set for the equations and the smooth properties of the operator semigroup are presented, leading to the existence of a global attractor.

• General Summary

• Abstract

In this paper, we consider the fractional  critical Schr\"odinger equation (FCSE)

where $u \in \dot H^s( \mathbb{R}^N)$,  $N\geq 4$,  $0<s<1$  and $2^{\ast}_{s}=\frac{2N}{N-2s}$ is the critical Sobolev exponent of order $s$.

By virtue of the variational method and the concentration compactness principle with  the  equivariant group action, we obtain some new type of non-radial, sign-changing solutions of (FCSE) in the energy space $\dot H^s(\mathbb{R}^N)$. The key component is that we take the equivariant group action to construct several  subspace of $\dot H^s(\mathbb{R}^N)$  with trivial intersection, then combine the concentration compactness argument in the Sobolev space with fractional order  to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of (FCSE) in $\dot H^s(\mathbb{R}^N)$.

• General Summary

• Abstract

In fluid mechanics and astrophysics, relativistic Euler equations can be used to describe the effects of special relativity which are an extension of the classical Euler equations. In this paper, we will consider the initial value problem of relativistic Euler equations in an initial bounded region of $\mathbb{R}^{N}$. If the initial velocity satisfies

$$\mathop{\max}_{\vec{x}_{0}\in \partial\Omega(0)}\sum_{i=1}^{N}v_{i}^{2}(0,\vec{x}_{0})<\frac{c^{2}A_{1}}{2},$$

where $A_{1}$ is a positive constant depend on some sufficiently large $T^{*}$, then we can get the non-global existence of the regular solution for the $N$-dimensional relativistic Euler equations.

• General Summary

• Abstract

We develop a locally mass-conservative enriched Petrov-Galerkin (EPG) method without any penalty term for the simulation of Darcy flow in fractured porous media. The discrete fracture model is applied to model the fractures as the lower dimensional fracture interfaces. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions in the fractures and the surrounding porous media. We propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method. The solution of the pressure can be decoupled into two steps with a standard CG method and a post-processing correction. The post-processing correction based on the bubble functions in the matrix and the fractures can be solved separately, which is useful for parallel computing. We derive a priori and a posteriori error estimates for the problem. Numerical examples are presented to illustrate the performance of the proposed method.

• General Summary

• Abstract

The semilinear wave equation with logarithmic and polynomial nonlinearities is considered in this paper. By adjusting and using potential well method, we attain the global-in-time existence and infinite time blowup solutions at subcritical initial energy level $E(0)<d$. Then using additional conditions on initial data, these results are enlarged at critical case $E(0)=d$ and arbitrarily positive case $E(0)>0$.

• General Summary

• Abstract

In this paper, we  are concerned with the minimal regularity of both the density and the velocity for the weak solutions keeping energy equality

in the isentropic compressible Navier-Stokes equations. The energy equality criteria without upper bound of the density  are established  for the first time. Our results imply that the lower integrability of the density $\rho$ means that more integrability of  the velocity $v$ or the gradient  of the velocity  $\nabla v$ are necessary for energy conservation of the  isentropic compressible fluid and the inverse is also true.

• General Summary

• Abstract

This paper is concerned with the global classical solution and the asymptotic behavior to a kind of linearly degenerate quasilinear hyperbolic system in several space variables. When the semilinear terms contain at least two  waves with different propagation speeds, we can prove that the system considered admits a global classical solution by the weighted energy estimate under the small and suitable decay assumptions on the initial data. Furthermore, we can show that the solution converges to a solution of the linearized system based on the decay property of the nonlinearties.

• General Summary