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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 $$Ce^{-\lambda t}$$.

##### MSC:
 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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