On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space

A complete discretization scheme for an ill-posed Cauchy problem for abstract firstorder linear differential equations with sectorial operators in a Banach space is validated. The scheme combines a time semidiscretization of the equations and a finite-dimensional approximation of the spaces and operators. Regularization properties of the scheme are established. Error estimates are obtained in the case of approximate initial data under various a priori assumptions concerning the solution.     

On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

In this paper,we study the stability of solutions of the Cauchy problem for 1-D compressible NarvierStokes equations with general initial data.The asymptotic limit of solution is found,under some conditions.The results in this paper imply the case that the limit function of solution as t →∞ is a viscous contact wave in the sense,which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero.As a by-product,the decay rates of the solution for the fast diffusion equations are also obtained.The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.     

Sewing method for weak solutions of second-order hyperbolic equation with variable domains of discontinuous unbounded operators

We prove the existence and uniqueness of global weak solutions on the entire interval for the Cauchy problem for hyperbolic differential-operator equations with time-discontinuous operators that have variable domains and satisfy certain matching conditions at the points of discontinuity. To this end, we develop a method of successive sewing of existing local weak solutions of Cauchy problems on the smoothness intervals of the operators. The sewing method is based on special energy inequalities, which imply the time continuity of local weak solutions in the main space and of their first derivatives in some negative spaces and hence the existence of the corresponding limit values at the points of discontinuity. These values, with regard for the matching conditions, are taken for the initial data on each successive interval.     

Study of magnetosonic solitary waves for the electron magnetohydrodynamics equations

Electron magnetohydrodynamics equations are derived with allowance for nonlinearity, dispersion, and dissipation caused by friction between the ions and electrons. These equations are transformed into a form convenient for the construction of a numerical scheme. The interaction of codirectional and oppositely directed magnetosonic solitary waves with no dissipation is computed. In the first case, the solitary waves are found to behave as solitons (i.e., their amplitudes after the interaction remain the same), while, in the second case, waves are emitted that lead to decreased amplitudes. The decay of a solitary wave due to dissipation is computed. In the case of weak dissipation, the solution is similar to that of the Riemann problem with a structure combining a discontinuity and a solitary wave. The decay of a solitary wave due to dispersion is also computed, in which case the solution can also be interpreted as one with a discontinuity. The decay of a solitary wave caused by the combined effect of dissipation and dispersion is analyzed.     

Asymptotic stability of contact waves for the compressible Navier–Stokes equations

In this article, we study the large-time asymptotic behaviour of contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and prove that it is nonlinearly stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small.     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 240–256, May, 2006.     

Properties of finite-difference schemes for singular integrodifferential equations of index 1

Systems of integrodifferential equations with a singular matrix multiplying the highest derivative of the unknown vector function are considered. An existence theorem is formulated, and a numerical solution method is proposed. The solutions to singular systems of integrodifferential equations are unstable with respect to small perturbations in the initial data. The influence of initial perturbations on the behavior of numerical processes is analyzed. It is shown that the finite-difference schemes proposed for the systems under study are self-regularizing.     

The Cauchy problem for semilinear hyperbolic equations with cross discontinuous data

This paper is devoted to study Cauchy problems of multidimensional semilinear strictly hyperbolic equations of second order with strongly singular initial data, where the derivatives of the initial data have discontinuity on two smooth curves transversally intersecting each other. The existence of the solution is proved, meanwhile, it is precisely discribed the flowery structure of the singularity of the solution.     

一维Chaplygin气体欧拉方程组中波的相互作用

研究一维Chaplygin气体欧拉方程组中波的相互作用.方程组的波包含接触间断和在密度变量以及内能变量上同时具有狄拉克函数的狄拉克激波.根据这些波的不同组合,问题被分成了7种情形.通过详细地构造每种情形的整体解,获得了各种波相互作用的完整结果.特别地,对于一类初值,两个接触间断相互作用后,产生了一个狄拉克激波;然而,对于另外一类初值,一个狄拉克激波与一个接触间断相互作用后,狄拉克激波消失.这些都是波相互作用中非常特别的现象.     

On the expansion of solutions to differential equations with discontinuous right-hand side in a series in initial data and parameters

The conditions under which the solutions of equations with discontinuous right-hand sides depend on the initial data and parameters analytically are investigated. A definition is introduced, which specifies this dependence in the case where a surface of discontinuity exists.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 715–717, May, 1993.     

Iteratively regularized methods for irregular nonlinear operator equations with a normally solvable derivative at the solution

A group of iteratively regularized methods of Gauss–Newton type for solving irregular nonlinear equations with smooth operators in a Hilbert space under the condition of normal solvability of the derivative of the operator at the solution is considered. A priori and a posteriori methods for termination of iterations are studied, and estimates of the accuracy of approximations obtained are found. It is shown that, in the case of a priori termination, the accuracy of the approximation is proportional to the error in the input data. Under certain additional conditions, the same estimate is established for a posterior termination from the residual principle. These results generalize known similar estimates for linear equations with a normally solvable operator.     

瞬态两相流的矢通量分裂解

本文用矢通量分裂法求解了瞬态两相流问题.首先导出了瞬态两相流控制方程的特征根,然后将矢通量分裂成二个子矢量,使每一个子矢量仅包含一种符号的特征值,再利用空间的一侧差分,得到差分方程并求解,最后将数值预测与实验结果进行了比较,其结果十分吻合.     

Multiple-scale Homogenization for Weakly Nonlinear Conservation Laws with Rapid Spatial Fluctuations

We consider hyperbolic conservation laws with rapid periodic spatial fluctuations and study initial value problems that correspond to small perturbations about a steady state. Weakly nonlinear solutions are computed asymptotically using multiple spatial and temporal scales to capture the homogenized solution as well as its long-term behavior. We show that the linear problem may be destabilized through interactions between two solution modes and the periodic structure. We also show that a discontinuity, either in the initial data or due to shock formation, introduces rapid spatial and temporal fluctuations to leading order in its zone of influence. The evolution equations we derive for the homogenized leading-order solution are more general than their counterparts for conservation laws having no rapid spatial variations. In particular, these equations may be diffusive for certain general flux vectors. Selected examples are solved numerically to substantiate the asymptotic results.     

Evolution of weak discontinuities in shallow water equations

In this paper, we determine the critical time, when a weak discontinuity in the shallow water equations culminates into a bore. Invariance group properties of the governing system of partial differential equations (PDEs), admitting Lie group of point transformations with commuting infinitesimal operators, are presented. Some appropriate canonical variables are characterized that transform equations at hand to an equivalent form, which admits non-constant solutions. The propagation of weak discontinuities is studied in the medium characterized by the particular solution of the governing system.     

On a new boundary-value problem for the Navier-Stokes system with discontinuity in the type of boundary conditions depending on time

An initial boundary-value problem for the nonlinear system of Navier-Stokes equations with timedependent discontinuity in the type of boundary conditions is proposed to describe the dynamics of a hurricane over solid ground. The tools for the solution of the stated problem are displayed in the case of both smooth and nonsmooth evolution of a hurricane.This paper continues the authors' work [1].     

Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time‐asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter ε tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations

We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves.It is shown that the unique solution to the Navier-Stokes equations exists for all time,and converges to the Riemann solution to the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks,as the viscosity vanishes.In contrast to previous related works,where either the composite wave is absent or the efects of initial layers are ignored,this gives the frst mathematical justifcation of this limit for the compressible isentropic Navier-Stokes equations in the presence of both composite wave and initial layers.Our method of proof consists of a scaling argument,the construction of the approximate solution and delicate energy estimates.     

Completely continuous and related multilinear operators

Completely continuous multilinear operators are defined and their properties investigated. This class of operators is shown to form a closed multi-ideal. Unlike the linear case, compact multilinear operators need not be completely continuous. The completely continuous maps are shown to be the closure of a subspace of the finite rank operators. Hilbert-Schmidt operators are also considered. An application to finding error bounds for solutions of multipower equations is presented.     

The limit equilibrium of an anisotropic medium under the general plasticity condition

A theory of the limit equilibrium of an anisotropic medium under the general plasticity condition in the plane strain state is developed. The proposed yield criterion (the limit equilibrium condition) is obtained by combining the von Mises–Hill yield criterion of an ideally plastic anisotropic material and Prandtl's limit equilibrium condition for a medium under the general plasticity law. It is shown that the problem is statically determinate, i.e., if the boundary conditions are specified in stresses, the stress state in plastic region can only be obtained using equilibrium equations. It is established that the equations describing the stress state are hyperbolic and have two families of characteristic curves that intersect at variable angles. In deriving the equations describing the velocity field, the material is assumed to be rigid plastic, and the associated law of flow is applied. It is shown that the equations for the velocities are also hyperbolic, and their characteristic curves are identical with those of the equations for stresses. However, the directions of the principal values of the stress and strain rate tensors are different due to the anisotropy of the material. The characteristic directions differ from the isotropic case in that the normal and tangential components of the stress tensor do not satisfy the limit conditions. It is established that the equations obtained allow of partial solutions, and in this case, at least one family of characteristic curves consists of straight lines. The conditions along the lines of discontinuity of the velocity are investigated, and it is shown that, as in the isotropic case, these are characteristic curves of the system of governing equations. In the anisotropic formulation, the well-known Rankine problem of the limit state of a ponderable layer is solved. From an analysis of the velocity field it is shown that plastic flow of the entire layer is possible only for a slope angle equal to the angle of internal friction. For slope angles less than the angle of internal friction, the solutions obtained are solutions of problems of the pressure of the medium on the retaining walls. The change in this pressure as a function of the parameters of anisotropy is investigated, and turns out to be significant.