Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   • A. Majda

   Pages 1-29

3. Smooth Solutions and the Equations of Incompressible Fluid Flow

   • A. Majda

   Pages 30-80

4. The Formation of Shock Waves in Smooth Solutions

   • A. Majda

   Pages 81-110

5. The Existence and Stability of Shock Fronts in Several Space Variables

   • A. Majda

   Pages 111-156

6. Back Matter

   Pages 157-159

   PDF

Conservation laws arise from the modeling of physical processes through the following three steps: 1) The appropriate physical balance laws are derived for m-phy- t cal quantities, ul""'~ with u = (ul' ... ,u ) and u(x,t) defined m for x = (xl""'~) E RN (N = 1,2, or 3), t > 0 and with the values m u(x,t) lying in an open subset, G, of R , the state space. The state space G arises because physical quantities such as the density or total energy should always be positive; thus the values of u are often con­ strained to an open set G. 2) The flux functions appearing in these balance laws are idealized through prescribed nonlinear functions, F.(u), mapping G into J j = 1, ..• ,N while source terms are defined by S(u,x,t) with S a given smooth function of these arguments with values in Rm. In parti- lar, the detailed microscopic effects of diffusion and dissipation are ignored. 3) A generalized version of the principle of virtual work is applied (see Antman [1]). The formal result of applying the three steps (1)-(3) is that the m physical quantities u define a weak solution of an m x m system of conservation laws, o I + N(Wt'u + r W ·F.(u) + W·S(u,x,t))dxdt (1.1) R xR j=l Xj J for all W E C~(RN x R+), W(x,t) E Rm.

• Erhaltungssatz
• Gasdynamik
• Kompressible Strömung
• Stosswelle
• Systems
• flow

• Department of Mathematics, University of California, Berkeley, USA

  A. Majda

Policies and ethics