z-logo
Premium
Erhaltungssätze und schwache lösungen in der gasdynamik
Author(s) -
Bruhn G.,
Wendland W.
Publication year - 1985
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670070133
Subject(s) - conservation law , mathematics , lemma (botany) , divergence (linguistics) , simple (philosophy) , pure mathematics , mathematical analysis , differentiable function , partial differential equation , ecology , linguistics , philosophy , poaceae , epistemology , biology
The fundamental laws of Gasdynamics can be formulated very naturally as conservation laws in the form of integral relations. This formulation includes not only continuously differentiable processes but also the very important discontinuous shocks. On the other side one has the tool of weak solutions of the differential equations of Gasdynamics due to P. D. Lax and several other authors. While the conservation laws of integral type are determined by Physics in an unique way the differential equations of Gasdynamics, even if written in divergence form, are not. Hence the question arises which form of the differential equations in the weak sense is the “correct” interpretation of the physical conservation laws. This paper tries to give an answer by investigating the connections between the two formulations. At first the integral equations of Gasdynamics are written in space‐time divergence form. Thus, independently from Gasdynamics, one has Haar's lemma stating that for each weak solution of a partial differential equation (in divergence form) a corresponding integral equation of conservation law type is valid for almost every family member, the family consisting of some simple domains like spheres or squares. Moreover the converse of Haar's lemma is also true. In this paper Haar's lemma is extended to a more general class of domains. This yields that both formulations of conservation laws are essentially equivalent. Additionally a divergence definition due to C. Müller is considered. As is shown by a simple example C. Müller's divergence concept leads to a more general class of solutions, not all of them being solutions of the corresponding conservation laws.

This content is not available in your region!

Continue researching here.

Having issues? You can contact us here