Five Open Problems in Compressible Mathematical Fluid Dynamics

The problems below are motivated by some of the works I did in the past. They deal with the dynamics of compressible fluids, either viscous or inviscid. They could be good questions for mathematicians having originality and technical strength. They are not worth a million dollars, even not a penny, but they are interesting in their own. 1 Global-in-time Cauchy problem for Euler–Fourier system Problem #1. To develop a global-in-time theory of the Cauchy problem for the 1-D Euler–Fourier system. Initial data would only be constrained by finite energy and entropy, and possibly by local boundedness of appropriate quantities, like ρ, 1 ρ ,v,θ, 1 θ . The complete Navier-Stokes-Fourier system for a viscous, heat-conducting fluid is ∂tρ+div(ρv) = 0, ∂t(ρv)+Div(ρv⊗ v)+∇p(ρ,e) = DivT , ∂t ( 1 2 ρ|v|2 +ρe ) +div ( 1 2 ρ|v|2 +ρe+ p(ρ,e) ) v = div(T v+κ(ρ,e)∇θ), where the viscous tensor is given by T := μ(ρ,e)(∇v+(∇v)T )+(ζ(ρ,e)−2μ(ρ,e))(divv)Id. Under appropriate assumptions about the pressure law (ρ,e) 7→ p and the diffusion coefficients μ,ζ and κ, the existence of global-in-time renormalized solutions to the Cauchy problem has been proved ∗UMPA, UMR CNRS–ENS Lyon # 5669. École Normale Supérieure de Lyon, 46, allée d’Italie, F–69364 Lyon, cedex 07.