On hydrodynamic limits of the Vlasov-Navier-Stokes system (Q6640522)

The book is devoted to the description of the long time behavior of the Vlasov-Navier-Stokes system in three high friction regimes. It is divided in seven chapters completed with an appendix.\N\NIn Chapter 1, which is the introduction, the authors consider the motion of a dispersed phase of small particles flowing in a surrounding incompressible homogeneous fluid, and the Vlasov-Navier-Stokes system: \(\partial_{t}u+(u\cdot \nabla_{x})u-\Delta_{x}u+\nabla_{x}p=j_{f}-\rho_{f}u\), \(div_{x}u=0\), \(\partial_{t}f+v\cdot \nabla_{x}f+div_{v}[f(u-v)]=0\), \((x,v)\in \mathbb{T}^{3}\times \mathbb{R}^{3}\), where \(\rho_{f}(t,x)=\int_{\mathbb{R}^{3}}f(t,x,v)dv\), \(j_{f}(t,x)=\int_{\mathbb{R}^{3}}vf(t,x,v)\), stand respectively for the density and momentum of the particles. The flat torus \(\mathbb{T}^{3}=\mathbb{T}^{3}/(2\pi \mathbb{Z})^{3}\) is equipped with the normalized Lebesgue measure, so that \(Leb(\mathbb{R}^{3})=1\). The initial conditions \(u\mid_{t=0}(x)=u^{0}(x)\), \(f\mid_{t=0}(x,v)=f^{0}(x,v)\) are added. The authors introduce the associated energy \(E(t)=\frac{1}{2} \int_{\mathbb{T}^{3}\times \mathbb{R}^{3}}\left\vert v\right\vert ^{2}f(t,x,v)dxdv+\frac{1}{2}\int_{\mathbb{T}^{3}}\left\vert u(t,x)\right\vert ^{2}dx\), and dissipation \(D(t)=\frac{1}{2}\int_{\mathbb{T} ^{3}}\left\vert \nabla_{x}u(t,x)\right\vert ^{2}dx+\frac{1}{2}\int_{\mathbb{T}^{3}\times \mathbb{R}^{3}}\left\vert v-u(t,x)\right\vert ^{2}f(t,x,v)dxdv\), so that the identity \(\frac{d}{dt}E+D=0\) holds true, at least formally. They consider three high friction regimes of the Vlasov-Navier-Stokes system: the light particle, light and fast particle and fine particle regimes. With an appropriate change of variables, the preceding system is written in dimensionless form as: \(\partial_{t}u+(u\cdot \nabla_{x})u-K\Delta_{x}u+\nabla_{x}p=C(j_{f}-\rho_{f}u)\), \(div_{x}u=0\), \(\partial_{t}f+Av\cdot \nabla_{x}f+Bdiv_{v}[f(\frac{1}{A}u-v]=0\), for some constants \(A,K,B\). The light particle regime corresponds to \(A=1\), \(B=\frac{1}{\varepsilon}\), \(C=1\), \(K=1\), the light and fast particle regime corresponds to: \(A=\frac{1}{\varepsilon ^{\alpha}}\), \(B=\frac{1}{\varepsilon}\), \(C=1\), \(K=1\), with \(\alpha >0\), and the fine particle regime corresponds to: \(A=1\), \(B=\frac{1}{\varepsilon}\), \(C=\frac{1}{\varepsilon}\), \(K=1\). The authors rewrite the above Vlasov-Navier-Stokes system according to these special regimes and they derive the formal high friction limits.\N\NChapter 2 starts with an analysis of the available results concerning this problem. They present the main tools for the description of the behavior as \(\varepsilon \rightarrow 0\) of solutions to: \(\partial_{t}u_{\varepsilon, \gamma, \sigma}+(u_{\varepsilon, \gamma, \sigma}\cdot \nabla_{x})u_{\varepsilon, \gamma, \sigma}-\Delta_{x}u_{\varepsilon, \gamma, \sigma}+\nabla_{x}p_{\varepsilon, \gamma, \sigma}=\frac{1}{\gamma} j_{f_{\varepsilon, \gamma, \sigma}}-\rho_{f_{\varepsilon, \gamma, \sigma}}u_{\varepsilon, \gamma, \sigma}\), \(div_{x}u_{\varepsilon, \gamma, \sigma}=0\), \(\partial_{t}f_{\varepsilon, \gamma, \sigma}+\frac{1}{\sigma}v\cdot \nabla_{x}f_{\varepsilon, \gamma, \sigma}+\frac{1}{\varepsilon} div_{v}[u_{\varepsilon, \gamma, \sigma}(\sigma uu_{\varepsilon, \gamma, \sigma}-v)]=0\), \(\rho_{f_{\varepsilon, \gamma, \sigma}}(t,x)=\int_{\mathbb{R}^{3}}f_{\varepsilon, \gamma, \sigma}(t,x,v)dv\), \(j_{f_{\varepsilon, \gamma, \sigma}}(t,x)=\frac{1}{\sigma}\int_{\mathbb{R} ^{3}}vf_{\varepsilon, \gamma, \sigma}(t,x,v)dv\), with \(\sigma =1\), \(\gamma =1\), for the light particle regime, \(\sigma =\varepsilon \alpha\), \(\gamma =1\) with \(\alpha >0\), for the light and fast particle regime, and \(\sigma =1\), \(\gamma =\varepsilon\), for the fine particle regime. The authors present the main tools they use for the derivation of their main results. They introduce the characteristic curves associated to the Vlasov equation and a modulated energy associated with the preceding system on which they derive a conditional exponential decay. They use results proved by \textit{D. Han-Kwan}, \textit{A. Moussa}, and \textit{I. Moyano} in [Arch. Ration. Mech. Anal. 236, No. 3, 1273--1323 (2020; Zbl 1436.35043)] to control the moments. They use \(L^{p}L^{p}\), \(p>3\), maximal estimates for the Stokes operator on \(\mathbb{T}^{3}\), and bootstrap arguments that they adapt for the three regimes.\N\NIn Chapter 3, the authors introduce the kinetic energy: \(E_{\varepsilon, \gamma, \sigma}(t)=\frac{1}{2}\left\Vert u_{\varepsilon, \gamma, \sigma}\right\Vert_{L^{2}(\mathbb{T}^{3})}^{2}+\frac{\varepsilon}{2\gamma \sigma ^{2}}\int_{\mathbb{T}^{3}\times \mathbb{R}^{3}}\left\vert v\right\vert ^{2}f_{\varepsilon, \gamma, \sigma}(t,x,v)dxdv\), and the dissipation: \(D_{\varepsilon, \gamma, \sigma}(t)=\frac{1}{2}\left\Vert \nabla_{x}u_{\varepsilon, \gamma, \sigma}\right\Vert_{L^{2}(\mathbb{T}^{3})}^{2}+ \frac{1}{\gamma}\int_{\mathbb{T}^{3}\times \mathbb{R}^{3}}\left\vert \frac{v}{\sigma}-u_{\varepsilon, \gamma, \sigma}(t,x)\right\vert ^{2}f_{\varepsilon, \gamma, \sigma}(t,x,v)dxdv\), associated to the problem presented in Chapter 2. Assuming that \(u_{\varepsilon, \gamma, \sigma}^{0}\in L_{div}^{2}(\mathbb{T}^{3})=\{U\in L^{2}(\mathbb{T}^{3})\), \(div_{x}U=0\}\), \(0\leq f_{\varepsilon, \gamma, \sigma}^{0}\in L^{1}\cap L^{\infty}(\mathbb{T}^{3}\times \mathbb{R}^{3})\), and \((x,v)\rightarrow f_{\varepsilon, \gamma, \sigma}^{0}(x,v)\left\vert v\right\vert ^{2}\in L^{1}(\mathbb{T}^{3}\times \mathbb{R}^{3})\), a global weak solution to the Vlasov-Navier-Stokes system with initial condition \((u_{\varepsilon, \gamma, \sigma}^{0},f_{\varepsilon, \gamma, \sigma}^{0})\) is a pair \((u_{\varepsilon, \gamma, \sigma},f_{\varepsilon, \gamma, \sigma})\) such that: the distribution function \(f_{\varepsilon, \gamma, \sigma}\in L_{loc}^{\infty}(\mathbb{R}_{+};L^{1}\cap L^{\infty}(\mathbb{T}^{3}\times \mathbb{R}^{3}))\) is a renormalized solution of the Vlasov equation, the fluid velocity \(u_{\varepsilon, \gamma, \sigma}\in L_{loc}^{\infty}(\mathbb{R}_{+};L^{2}(\mathbb{T}^{3}))\cap L_{loc}^{2}(\mathbb{R}_{+};H^{1}(\mathbb{T}^{3}))\) is a Leray solution to the Navier-Stokes equations, \(j_{\varepsilon, \gamma, \sigma}-\rho_{\varepsilon, \gamma, \sigma}u\in L_{loc}^{2}(\mathbb{R}_{+};H^{-1}(\mathbb{T}^{3}))\), for almost all \(t\geq s\geq 0\) (including \(s=0\)), and the energy-dissipation inequality: \(E_{\varepsilon, \gamma, \sigma}(t)+\int_{s}^{t}D_{\varepsilon, \gamma, \sigma}(\tau)d\tau \leq E_{\varepsilon, \gamma, \sigma}(s)\) holds. They recall conservation laws for the Vlasov equation with friction, a higher regularity estimate for the Navier-Stokes equations, a rough estimate on the Brinkman force \(F_{\varepsilon, \gamma, \sigma}=\frac{1}{\gamma} (j_{\varepsilon, \gamma, \sigma}-\rho_{\varepsilon, \gamma, \sigma}u_{\varepsilon, \gamma, \sigma})\). They introduce the modulated energy dissipation and the Chapter ends with an initiation to the bootstrap argument that they will use to determine strong existence times.\N\NChapter 4 is devoted to the analysis of the light and light and fast particle regimes. The authors first prove precise estimates on the Brinkman force, thanks to a desingularization with respect to \(\varepsilon\), based on the Lagrangian structure of the Vlasov equation. They also complete the bootstrap argument in these cases, using an \(L^{p}\) parabolic maximal estimate combined with the exponential decay of the modulated energy through an interpolation argument. They conclude the proof of the bootstrap argument, using using an \(L^{\infty}L^{2}-L^{2}\dot{H}^{1}\) energy estimate for the Navier-Stokes part and Wasserstein stability estimates for the Vlasov part. They first proof the convergence results in the mildly well-prepared and well-prepared cases, then in the general case. They derive the long time behavior of solutions to the Vlasov-Navier-Stokes system in the light and light and fast particle regimes.\N\NIn Chapter 5, the authors analyze the fine particle regime. They introduce relevant higher dissipation functionals. They derive the desingularization of the Brinkman force and prove uniform \(L^{p}\) (in time and space) bounds. They conclude the verification of the bootstrap argument in the case of mildly well-prepared initial data and they prove the convergence result in this case. They prove relative entropy estimates and they prove the convergence result in the well-prepared case.\N\NIn Chapter 6, the authors come back to the light and light and fast particle regimes. They prove uniform \(L^{\frac{p}{p-}}1L^{p}\) estimates for the convective term and the Brinkman force in the Navier-Stokes equation, they conclude the bootstrap argument and obtain the convergence towards smooth solutions of the transport-Navier-Stokes equation, in the mildly well-prepared and well-prepared, then cases. They finally introduce higher dissipation and relative entropy in the light and light and fast particle regimes.\N\NChapter 7 presents some further developments, first in the 2D case, then for mixtures of two types of particles with different radii. They derive a Boussinesq-Navier-Stokes type system from the Vlasov-Navier-Stokes system, and the Chapter ends with some open problems.\N\NIn the Appendix, the authors recall properties of the Sobolev and Besov (semi-)norms and Wasserstein-1 distance, the Gagliardo-Nirenberg interpolation estimates, estimates for smooth solutions to the incompressible Navier-Stokes equations, and maximal parabolic regularity estimates for the Stokes equation.\N\NThe book presents an up-to-date analysis of the Vlasov-Navier-Stokes system and of the long time behavior of solutions in some special regimes. The proofs are given with details.