Global smooth solutions of Euler equations for Van der Waals gases

Global smooth solutions of Euler equations for Van der Waals gases Magali Lécureux-Mercier? February 17, 2011 Abstract We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in Hm, for m large enough. To do so, we introduce a specific symmetrization allowing areas of null density. Next, we make estimates in Hm, using for some terms the estimates done by Grassin, who proved the same theorem in the easier case of a perfect polytropic gas. We treat the remaining terms separately, due to their nonlinearity. 2000 Mathematics Subject Classification: 35L60, 35Q31, 76N10. Keywords: Euler compressible equations, smooth solutions, special symmetrization. 1 Introduction We are interested in the Cauchy problem for Euler compressible equations, describing the evolution of a gas whose thermodynamical and kinetic properties are known at time t = 0. More specifically, we are concerned with the life span of smooth solutions. Various authors, in particular Sideris [20, 21], Makino, Ukai and Kawashima [13] and Chemin [2, 3] have given criteria for mathematical explosion. We know also that there exist global in time solutions for well-chosen initial data. Li [12], Serre [19] and Grassin [7] prove, for example, the global in time existence of regular solutions under some hypotheses of “expansivity”.

• there exist global


• ?? ?


• global classical


• law coincides


• euler compressible


• can define


• gases


• state law


• euler equations