In this paper, we study the global exisitence of $L^\infty$ weak entropy solution to the initial boundary value problem for compressible Euler equations with relaxtion and the large time asymptotic behavior of the solution. Motivated by the sub-characterisitic conditions, we proposed some structural conditions on the relaxation term comparing with the pressure function. These conditions are proved to be sufficient to construct the global $L^\infty$ entropy weak solution and to prove the equilibrium state is the global attactor of all physical weak solutions. Furthermore, the convergence rate is proved to be exponential in time. The proof is based on the entropy dissipation principle.

We consider an initial-boundary value problem for the equations of one-dimensional motions of a compressible viscous heat-conducting gas coupled with radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients, we prove that the problem admits a unique weak solution.

The present paper considers, in a linearized context, the interaction between two gasdynamic objects: a turbulence model and, respectively, a planar shock discontinuity. The incident turbulence, regarded as a perturbation, is modelled by a nonstatistical/noncorrelative superposition of some finite (or point core) planar vortices. $\bullet$ The linearized (with shock) context assumes a minimal nonlinearity. It considers a linearized problem: a linear problem with a nonlinear subconscious (in the sense of P.D. Lax and A. Majda; see $\S$2). The resultant perturbation is regarded as a solution (``interaction solution") of such a linearized problem. $\bullet$ In presence of a nonlinear subconscious the interaction solution is essentially constructed as an admissible solution.

Unimprovable, in a certain sense, sufficient conditions of solvability and unsolvability of nonlocal problems are found for the differential system

$$ \frac{dx_i}{dt} =f_i(t,x_1,\dots,x_n) \quad (i=1,\dots,n), $$

where each of the functions $f_i:[a,b]\times R^n \to R$ $(i=1,\dots,n)$ may be superlinear or sublinear with respect to phase variables.

In this paper, we propose an alternative direct algebraic method of constructing, for nonlinear evolution partial differential equations, conservation laws that depend not only on dependent variables and its derivatives but also explicitly on independent variables. As illustration, the fifth order Korteweg de Vries (fKdV) and modified $(n+1)$-dimensional Zakharov-Kuznetvov (ZK) equations are probed.

The mathematics of time scales has recently received much attention and holds great promise in a number of areas. In this paper we propose a new area of mathematics, namely the theory of stochastic dynamic equations, which unifies the theories of stochastic differential and difference equations. We give an example involving stochastic dynamic equations, namely an equation modeling a stock price.

The representation of a high dimensional machine learning (ML) feature space $F$ as a function space for the purpose of denoising data is introduced. We illustrate an application of such a representation of feature vectors by applying a local averaging denoising method for functions on Euclidean and metric spaces (together with its graph generalization) to the regularization of feature vectors in ML. We first discuss this technique for noisy functions on $\mathbb{R}$, and then extend it to functions defined on graphs and networks. This method exhibits a paradoxical property of the bias-variance problem in machine learning, namely, that as the scale over which averages are taken decreases, the error rate for classification first decreases and then increases. This approach is tested on two benchmark DNA microarray data sets used for classification of breast tumors based on predicted metastasis.