By Etienne Emmrich, Petra Wittbold

This article incorporates a sequence of self-contained studies at the cutting-edge in several components of partial differential equations, awarded by way of French mathematicians. issues contain qualitative houses of reaction-diffusion equations, multiscale equipment coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation laws.

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The reason is, the equations we study model nonlinear physical phenomena (called “processes” in the sequel) which are time-irreversible, and the function which characterizes this irreversibility is called “entropy”. 1) is, certainly, the simplest model for the displacement of a gas in a tube; in more correct (more precise) models, also the pressure of the gas is present, moreover, the density of the gas enters the equations when the gas is compressible. The entropy function S is expressed with the help of the two latter quantities characterizing the gas, namely the pressure and the density.

First, let us see which rays can separate two smoothness components of such a solution: two adjacent components may correspond either to two different constant states, or to a constant state and to the restriction of the function x/t on some cone with the vertex (0, 0). 5) that two constant functions u(t, x) ≡ u1 and u(t, x) ≡ u2 , ui = const, can only be juxtaposed along the ray x= f (u2 ) − f (u1 ) 1 u22 − u21 u2 + u1 t= t= t, u2 − u1 2 u2 − u1 2 and because of the entropy increase condition, the jump is admissible only when u jumps from a greater to a smaller value (we mean that the direction of the jump is such that x grows).

Indeed the convexity (respectively, the concavity) of a function means, by definition, that the chord joining two arbitrary points of the graph of the function lies above (respectively, lies below) the graph itself. 3) and, in addition, the admissibility condition formulated in the present section. 1. 7. 6 Where does this name come from? The reason is, the equations we study model nonlinear physical phenomena (called “processes” in the sequel) which are time-irreversible, and the function which characterizes this irreversibility is called “entropy”.