摘要:

AbstractIn this paper the existence and uniqueness of solutions of the following initial boundary value problem for non‐linear symmetric hyperbolic equations of the first order\documentclass{article}\pagestyle{empty}\begin{document}$$ {\bf{E}}\left({t,{\bf{x}},{\bf{u}}} \right){\bf{u}}_t + {\bf{A}}_i \left({t,{\bf{x}},{\bf{u}}} \right){\bf{u}}_{x_i } + {\bf{B}}\left({t,{\bf{x}},{\bf{u}}} \right){\bf{u}} = {\bf{F}}\left({t,{\bf{x}}} \right), $$\end{document}\documentclass{article}\pagestyle{empty}\begin{document}$$ {\bf{u}}|_{t = 0} = {\bf{u}}_0 \left({\bf{x}} \right), $$\end{document}\documentclass{article}\pagestyle{empty}\begin{document}$$ {\bf{M}}\left({t,{\bf{x}},{\bf{u}}} \right){\bf{u}}|_{\partial \Omega } = {\bf{g}}\left({{\bf{x}}\prime,t} \right), $$\end{document}are shown, whereM=I+−S, has the same from as the Kreiss' condition, butSmust be sufficiently small (I+is the unit matrix in the space generated by eigenvectors of the matrix −A·n̄, corresponding to positive eigenvalues) andn̄is a unit outward vector normal to the boundary. The main result of the paper is obtaining ana prioriestimate for non‐linear equations. This estimate is obtained for sufficiently small time and norms of given data functions. The existence of solutions is proved by the method of successive approximations, which can be used because at each step such properties as symmetry of matrices and the numbers of positive and negative eigenvalues of the matrix −A·n̄are assured. This can be done because we restrict our attention to such systems of equations for which these properties are satisfied for solutions from some neighbourhood of initial datau0. Therefore, using the fact that solutions in the class of continuous functions are sought, these properties can be satisfied for sufficiently small time. Moreover, some examples of initial boundary value problems for equations of hydrodynamics and magnetohydrodynamics

ISSN:0170-4214    

DOI:10.1002/mma.1670110201

出版商:John Wiley&Sons, Ltd    

年代:1989

数据来源: WILEY

摘要:

AbstractWe hereafter propose and analyse a discontinuous finite element method for a plane stress Hencky problem. For that purpose we begin by proving an existence result for the continuous problem. A kind of Green's formula between\documentclass{article}\pagestyle{empty}\begin{document}$$ BD\left(\Omega \right) = \left\{{u \in {\rm{L}}^1 \left(\Omega \right),\varepsilon _{ij} (u) \in M_1 \left(\Omega \right)} \right\}{\rm{and}}H\left(\Omega \right) = \left\{{\sigma \in L^\infty \left(\Omega \right),div\sigma \in {\rm{L}}^2 \left(\Omega \right)} \right\} $$\end{document}and other intermediate results that may be of independent interest are presented and established separately. Then we formulate the discretized problem, give an existence result for it and prove a result of weak convergence of a subsequence of discrete solutions to a solution of the continuous problem.

ISSN:0170-4214    

DOI:10.1002/mma.1670110202

出版商:John Wiley&Sons, Ltd    

年代:1989

数据来源: WILEY

摘要:

AbstractWe consider a Helmholtz equation in a number of Lipschitz domains inn≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are describe

ISSN:0170-4214    

DOI:10.1002/mma.1670110203

出版商:John Wiley&Sons, Ltd    

年代:1989

数据来源: WILEY

ISSN:0170-4214    

DOI:10.1002/mma.1670110204

出版商:John Wiley&Sons, Ltd    

年代:1989

数据来源: WILEY

摘要:

AbstractWe present the asymptotic analysis of a quasilinear hyperbolic–hyperbolic singular perturbation problem in one dimension. The leading part of the analysis concerns the construction of some shock layers associated with discontinuities of a hyperbolic problem. This study is a generalization of the case of viscous perturbation for a hyperbolic proble

ISSN:0170-4214    

DOI:10.1002/mma.1670110205

出版商:John Wiley&Sons, Ltd    

年代:1989

数据来源: WILEY

摘要:

AbstractA system of quasi‐linear first‐order equations written in the divergence form and constrained by the unilateral differential inequality (the second law of thermodynamics) with a strictly concave entropy function is analysed. In the class BV, i.e. a subset of regular distributions represented by functions of bounded variation in the sense of Tonelli‐Cesari, a weak solution to the system is defined. The parabolized version of the system is also discussed in order to define an admissible weak solution as a limit of a sequence of Lipschitz continuous solutions to the parabolic problem. It is proved that an admissible weak solution of the Cauchy problem is unique in the cla

ISSN:0170-4214    

DOI:10.1002/mma.1670110206

出版商:John Wiley&Sons, Ltd    

年代:1989

数据来源: WILEY

摘要:

AbstractIn the present paper sufficient conditions for stability of the solutions of linear systems of differential equations with variable structure and impulse effect are found.

ISSN:0170-4214    

DOI:10.1002/mma.1670110207

出版商:John Wiley&Sons, Ltd    

年代:1989

数据来源: WILEY