Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows
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Calculus of Variations
Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows Yanmin Mu1,2 · Dehua Wang3 Received: 30 October 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we study the three-dimensional non-isentropic compressible fluid–particle flows. The system involves coupling between the Vlasov–Fokker–Planck equation and the non-isentropic compressible Navier–Stokes equations through momentum and energy exchanges. For the initial data near the given equilibrium we prove the global well-posedness of strong solutions and obtain the optimal algebraic rate of convergence in the threedimensional whole space. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential. New ideas and techniques are developed to establish the well-posedness and large-time behavior. For the global well-posedness our methods are based on the new macro–micro decomposition which involves less dependence on the spectrum of the linear Fokker–Plank operator and fine energy estimates; while the proofs of the optimal large-time behavior rely on the Fourier analysis of the linearized Cauchy problem and the energy-spectrum method, where we provide some new techniques to deal with the nonlinear terms. Mathematics Subject Classification 35Q30 · 76D03 · 76D05 · 76D07
1 Introduction In this paper we study the global well-posedness and large time behavior of strong solutions for the three-dimensional fluid–particle flows, governed by the following Navier–Stokes
Communicated by P. Rabinowitz.
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Dehua Wang [email protected] Yanmin Mu [email protected]
1
School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210046, China
2
School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing 210023, China
3
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA 0123456789().: V,-vol
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Y. Mu, D. Wang
equations of compressible non-isentropic fluids coupled with the Vlasov–Fokker–Planck equation of particles [3,13,33]: ∂t n + ∇ · (nu) = 0,
(1.1)
∂t (nu) + ∇ · (nu ⊗ u) − μu + ∇ p = M, ∂t (n E) + ∇ · (n E + p)u − κθ˜ = F ,
(1.2)
∂t F + v · ∇x F = L u,θ˜ F,
(1.4)
(1.3)
where, n = n(t, x) ≥ 0, u = u(t, x) ∈ R3 , p = p(t, x) ≥ 0, E = E(t, x) ≥ 0, θ˜ = θ˜ (t, x) ≥ 0 for (t, x) ∈ R+ × denote the density, velocity, pressure, total energy, and temperature of the fluids, respectively; F = F(t, x, v) ≥ 0 for (t, x, v) ∈ R+ × × R3 denotes the density distribution function of particles in the phase space; the spatial domain is = R3 or T3 (a periodic domain in R3 ); and μ, κ are the viscosity and heat conductivity constants. The total energy E, internal energy e, pressure p, and temperature θ˜ satisfy the p , where γ > 1 is the adiabatic following relations: E = e + 21 |u|2 , p = Rn θ˜ , e = (γ −1)n constant and R > 0 is constant. The Fokker–Planck opera
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