Weak and strong solutions of equations of compressible magnetohydrodynamics

International audienceThis article proposes a review of the analysis of the system of magnetohydrodynamics (MHD). First, we give an account of the modelling asumptions. Then, the results of existence of weak solutions, using the notion of renormalized solutions. Then, existence of strong solutions in the neighbourhood of equilibrium states is reviewed, in particular with the method of Kawashima and Shizuta. Finally, the special case of dimension one is highlighted : the use of Lagrangian coordinates gives a simpler system, which is solved by standard techniques

Scale Analysis of Compressible Flows from an Application Perspective

From an application perspective the fluid flow equations are studied to gain insight into particular solutions related to concrete flow problems. Methods of asymptotic – or scale – analysis provide powerful means to this end. By exploiting the smallness of characteristic dimensionless parameters in a given flow regime, these methods lead to approximate reduced equations in a systematic fashion. Comparison of the reduced and full models yields new insights in that it reveals which mechanisms are and which are not important in the considered flow regime. Moreover, the reduced equations are often easier to solve numerically or, in the best of all cases, even admit analytical solution. Asymptotics is sometimes considered a “dark art”: From a mathematician’s perspective, there is too much emphasis on physics-based and formal arguments rather than on rigorous statements and proofs. For the applied scientist or engineer, in turn, asymptotic techniques involve steps and procedures that seem to satisfy mathematical needs only while being hard to justify in physical terms. This contribution is an attempt at addressing some of these concerns by walking through a few established examples of multiple scales analysis and by carefully explaining each step on the way