Chaotic mixing of distinct fluids produces a convoluted structure to the interface separating these fluids. For miscible fluids (as considered here), this interface is defined as a $50\%$ mass concentration isosurface. For shock wave induced (Richtmyer-Meshkov) instabilities, we find the interface to be increasingly complex as the computational mesh is refined. This interfacial chaos is cut off by viscosity, or by the computational mesh if the Kolmogorov scale is small relative to the mesh. In a regime of converged interface statistics, we then examine mixing, i.e. concentration statistics, regularized by mass diffusion. For Schmidt numbers significantly larger than unity, typical of a liquid or dense plasma, additional mesh refinement is normally needed to overcome numerical mass diffusion and to achieve a converged solution of the mixing problem. However, with the benefit of front tracking and with an algorithm that allows limited interface diffusion, we can assure convergence uniformly in the Schmidt number. We show that different solutions result from variation of the Schmidt number. We propose subgrid viscosity and mass diffusion parameterizations which might allow converged solutions at realistic grid levels.
It is shown that self-similar $BV$ solutions of genuinely nonlinear strictly hyperbolic systems of conservation laws are special functions of bounded variation, with vanishing Cantor part.
This paper is devoted to the study of the proper setting of the boundary conditions for the boundary value problems of the hyperbolic-elliptic coupled systems of first order. The wellposedness of the corresponding boundary value problems is also established. The Lopatinski conditions for the boundary value problems of the elliptic systems is then extended to the case for hyperbolic-elliptic coupled systems. The result in this paper can be applied to the Euler system in fluid dynamics, especially to give wellposed boundary value problems describing subsonic flow.
In this paper, we propose a GL method for solving the ordinary and the partial differential equation in mathematical physics and chemics and engineering. These equations govern the acustic, heat, electromagnetic, elastic, plastic, flow, and quantum etc. macro and micro wave field in time domain and frequency domain. The space domain of the differential equation is infinite domain which includes a finite inhomogeneous domain. The inhomogeneous domain is divided into finite sub domains. We present the solution of the differential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains. Actualy, we propose an explicit representation of the inhomogeneous parameter nonlinear inversion. The analytical solution of the equation in the infinite homogeneous domain is called as an initial global field. The global field is updated by local scattering field successively subdomain by subdomain. Once all subdomains are scattered and the updating process is finished in all the sub domains, the solution of the equation is obtained. We call our method as Global and Local field method, in short , GL method. It is different from FEM method, the GL method directly assemble inverse matrix and gets solution. There is no big matrix equation needs to solve in the GL method. There is no needed artificial boundary and no absorption boundary condition for infinite domain in the GL method. We proved several theorems on relationships between the field solution and Green's function that is the theoretical base of our GL method. The numerical discretization of the GL method is presented. We proved that the numerical solution of the GL method convergence to the exact solution when the size of the sub domain is going to zero. The error estimation of the GL method for solving wave equation is presented. The simulations show that the GL method is accurate, fast, and stable for solving elliptic, parabolic, and hyperbolic equations. The GL method has advantages and wide applications in the 3D electromagnetic (EM) field, 3D elastic and plastic etc seismic field, acoustic field, flow field, and quantum field. The GL method software for the above 3D EM etc field are developed.
In this paper, we study three special families of strong entropy-entropy flux pairs $(\eta_{0},q_{0}), (\eta_{\pm},q_{\pm})$, represented by different kernels, of the isentropic gas dynamics system with the adiabatic exponent $\gamma \in (3, \infty)$. Through the perturbation technique through the perturbation technique, we proved, we proved the $H^{-1}$ compactness of $\eta_{it}+q_{ix}, i=1,2,3$ with respect to the perturbation solutions given by the Cauchy problem (6) and (7), where $(\eta_{i},q_{i})$ are suitable linear combinations of $(\eta_{0},q_{0}), (\eta_{\pm}, q_{\pm})$.
This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.
The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.
We investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev nonlinearity and a term of lower order. We allow a coefficient of u in equation (1.1) to be unbounded. We prove the existence of a solution in a weighted Sobolev space.
In this paper, we consider the existence of multiple solutions for a class of singular nonlinear boundary value problem involving critical exponent in Weighted Sobolev Spaces. The existence of two solutions is established by using the Ekeland Variational Principle. Meanwhile, the uniqueness of positive solution for the same problem is also obtained under different assumptions.
Consider the following Neumann problem $$ d\Delta u - u + k(x)u^p=0 \ {\rm and } \ u>0 \ {\rm in} \ B_1, \ \ \small\frac{\partial u}{\partial\nu}=0\ {\rm on} \ \partial B_1 , \tag $\ast$ $$ where $d>0$, $B_1$ is the unit ball in ${\Bbb{R}^N}$, $k(x)=k(|x|)\not \equiv 0$ is nonnegative and in $C(\overline B_1)$, $1<p<\frac{N+2}{N-2}$ with $ N\geq 3$. It was shown in [2] that, for any $d>0$, problem $(*)$ has no nonconstant radially symmetric least energy solution if $k(x)\equiv 1$. By an implicit function theorem we prove that there is $d_0>0$ such that $(\ast)$ has a unique radially symmetric least energy solution if $d>d_0$, this solution is constant if $k(x)\equiv 1$ and nonconstant if $k(x)\not\equiv 1$. In particular, for $k(x)\equiv 1$, $d_0$ can be expressed explicitly.
In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed. Particularly, the $2p+1$-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.
By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball $B^{N}$ and on the boundary $S^{N-1}$ the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to $0^+$ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.
In this paper, we are concerned with the elliptic system of $$ \cases-\Delta u+V(x)u= g(x,v), \qquad & x \in{\mathbf R}^N,\\ -\Delta v+V(x)v= f(x,u),& x \in{\mathbf R}^N, \endcases $$ where $V(x)$ is a continuous potential well, $f, g$ are continuous and asymptotically linear as $t\rightarrow \infty$. The existence of a positive solution and ground state solution are established via variational methods.
In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for $n\times n$ hyperbolic system of conservation laws with artificial viscosity in the half line $(0,\infty)$. We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.