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Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. (English) Zbl 1215.35127
The paper proves a global well-posedness result for a two-phase flow system modelling a fluid-particle interaction. The authors consider a Vlasov-Fokker-Planck equation for the particles \(\partial_t F+\xi\cdot \nabla_\xi F= \nabla_\xi\cdot ((\xi-u)F+\nabla_\xi F)\), coupled to an incompressible Euler equation for the fluid \(\partial_t u+u\cdot\nabla_x u+\nabla_x p=\int_{\mathbb R^3}(\xi-u)\,d\xi \). Here \(F\) is the density distribution of particles, \(u\) is the velocity field which satisfies the incompressibility condition \(\nabla_x\cdot u=0\) and \(p\) is the pressure of the fluid. Initial conditions \(F(0,x,\xi)=F_0(x,\xi)\) and \(u(0,x)=u_0(x)\) are prescribed, \(u_{0}\) being divergence-free. Introducing the change of unknown density \(F=M+M^{1/2}f\), where \(M(\xi)=\exp (-|\xi|^2/2)/(2\pi)^{3/2}\) the authors obtain a simplified coupled Cauchy problem for \((f,u)\). Assuming that \(F_0\geq 0\) and that \(\|f_0\|_{L_\xi^2(H_\xi^3)}+\|u_0\|_{H^3}\) is small enough, the authors prove the existence of a unique global solution \((f,u)\) of this Cauchy problem. Under further hypotheses on the initial conditions, this solution satisfies a time-decay estimate \(C(1+t)^{-3/4+\varepsilon }\). The proof of this existence result is mainly based on new energy estimates and on an adapted version of Kawashima’s hyperbolic-parabolic dissipation estimates. The paper ends with the study of the Cauchy problem now posed in a 3D torus \(\mathbb T^3\). The authors here prove an exponential time-decay estimate as \(Ce^{-\lambda t}\).

35Q35 PDEs in connection with fluid mechanics
35Q31 Euler equations
35Q84 Fokker-Planck equations
76T20 Suspensions
35B40 Asymptotic behavior of solutions to PDEs
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