Global Smooth Solutions in R3 to Short Wave-Long Wave Interactions Systems for Viscous Compressible Fluids

The short wave-long wave interactions for viscous compressible heat conductive fluids is modeled, following Dias & Frid (2011), by a Benney-type system coupling Navier-Stokes equations with a nonlinear Schrödinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R3 when the initial data are small smooth perturbations of an equilibrium state. This is a joint work with Roghua Pan and Weizhe Zhang. Recently, this analysis has been extended to the magnetohydrodynamics equations, in a joint work with Ronghua Pan and Junxiong Xia, currently in preparation. Hermano Frid Benney-type interactions systems for compressible fluids The model The system describing short wave-long wave interactions for compressible fluids with which we are concerned reads as follows ρt + div (ρu) = 0, (ρu)t + div (ρu ⊗ u) = μ∆u + ν∇div u −∇p(ρ, θ) + α∇(g ′( 1 ρ )h(|w ◦ Y |)), θt + u · ∇θ + θpθ cθρ div u = 1 cθρ (κ∆θ + Ψ), i ∂w ∂t + ∆y w = |w |w + α̃g(v)h′(|w |) w , (1) whose terms will be explained subsequently, and for which it is the main purpose of this paper to solve the Cauchy problem in R, so (t, x) ∈ [0,∞)× R, for prescribed initial data (ρ(0, x), u(0, x), θ(0, x),w(0, y)) = (ρ0(x), u0(x), θ0(x),w0(y)), (2) which are small and smooth perturbations of a constant state, say, ρ = ρ̄0, u = 0, θ = θ̄0, w = 0. Hermano Frid Benney-type interactions systems for compressible fluids