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Mechanisms of Lagrangian analyticity in fluids. (English) Zbl 1422.76019

In this long paper, the author elucidates mechanisms which lead to an analyticity property of the Lagrangian trajectories associated to the dynamics of inviscid fluids and he illustrates these mechanisms on different systems. In Section 2, the author considers a general ODE \(\frac{dX}{dt}=F(X)\) with the initial data \(X| _{t=0}=\mathrm{Id}\), where \(F\) is an operator defined on a set of maps from \(\mathbb{R}^{d}\) to \(\mathbb{C} ^{d}\), depending on an initial data \(u| _{t=0}=u_{0}\). The author presents properties of \(F\) (local boundedness and preservation of analyticity) which ensure the analyticity of \(X\) with respect to time. In Section 3, the author considers the case of incompressible Euler equations on \(\mathbb{R}^{2}\): \( \frac{\partial u}{\partial t}+u\cdot \nabla u+\nabla p=0\) with the initial data \(u| _{t=0}=u_{0}\). He observes that the vorticity \(\omega =\partial _{1}u_{2}-\partial _{2}u_{1}\) satisfies the transport equation \(\frac{ \partial \omega }{\partial t}+u\cdot \nabla \omega =0\) and he obtains the equation for the trajectories \[ \frac{dX}{dt}(\alpha ,t)=\frac{1}{2\pi }\int \frac{(X(\alpha ,t)-X(\beta ,t))^{\perp }}{\left\vert X(\alpha ,t)-X(\beta ,t)\right\vert ^{2}}\omega _{0}(\beta )\,d\beta, \] where \(\omega _{0}\) is the initial vorticity and \(y^{\perp }=(-y_{2},y_{1})\). The main result of this section proves an analyticity property of the vortex patch solution to this equation, assuming that the initial vorticity is given as \(\omega _{0}(\alpha )=1\) if \(\alpha \in \Omega \) an open subset of \(\mathbb{R}^{2}\) whose boundary is a finite union of non-intersecting closed \(C^{2}\) curves and \( \omega _{0}(\alpha )=0\) otherwise. The author proves that the function \(F\) associated to the operator \(K=\frac{1}{2\pi }\frac{(-y_{2}-y_{1})}{ y_{1}^{2}+y_{2}^{2}}\) satisfies the properties introduced in Section 2. In Section 4, the author considers the Euler-Poisson system in \( \mathbb{R}^{3}\): \[ \begin{aligned} \partial _{t}\rho +u\cdot \nabla \rho &=-\rho\operatorname{div}u,\\ \partial _{t}u+u\cdot \nabla u &= q\nabla \Delta ^{-1}\rho, \end{aligned} \] with the initial condition \((\rho ,u)=(\rho _{0},u_{0})\) at \(t=0\), where \(q=\pm 1\). The Lagrangian trajectory here satisfies \(\frac{d^{2}X}{dt^{2}}(\alpha ,t)=\int K(X(\alpha ,t)-X(\beta ,t))\rho _{0}(\beta )\,d\beta \), where \(K(y)=\frac{1}{ 4\pi }\frac{y}{\left\vert y\right\vert ^{3}}\) which can be written as a system of first-order ODEs: \[ \frac{dX}{dt}(\alpha ,t)=V(\alpha ,t), \quad \frac{ dV}{dt}(\alpha ,t)=\int K(X(\alpha ,t)-X(\beta ,t))\rho _{0}(\beta )\,d\beta. \] The author here assumes that \(\rho _{0}\in C_{c}^{s}\) and \(u_{0}\in H^{s}\) for \(s\geq 6\). The main result proves a local existence result for \((\rho ,u) \) and that the trajectory \(X(\alpha ,t)\) is analytic in \(t\) at \(t=0\) for every \(\alpha \in \mathbb{R}^{3}\). Once the function \(F\) has been defined, the proof verifies its properties which ensure the analyticity of the trajectory. In Section 5, the author considers the 2D compressible Euler equation \[ \partial _{t}\rho +\operatorname{div}(\rho u)=0, \quad \partial _{t}u+u\cdot \nabla u+\frac{1}{\rho }\nabla p=0. \] The author here proves that, even if the initial data \((\rho _{0},u_{0})\) are smooth and compactly supported, there exist Lagrangian trajectories which are not analytic in a small disc about \(t=0\). In the final Section 6, the author considers the Vlasov-Poisson system \[ \partial _{t}f(x,v,t)+v\cdot \nabla _{x}f(x,v,t)+E(x,t)\cdot \nabla _{v}f(c,v,t)=0, \] where \(E(x,t)=-\nabla _{x}\phi (x,t)\), \(\phi \) being the solution of \[ -\Delta _{x}\phi (x,t)=q\rho (x,t)=q\int f(x,w,t)\,dw. \] An initial density \(f\mid _{t=0}=f_{0}\in \mathcal{S}(\mathbb{R}^{4})\) is given. The author proves that the Lagrangian trajectory is the solution of the system \[ \frac{dX}{dt}(\zeta ,t)=V(\zeta ,t), \quad \frac{dV}{dt}(\zeta ,t)=\int K(X(\zeta ,t)-X(\Xi ,t))f_{0}(\Xi )\,d\Xi, \] with the initial condition \((X,V)(\zeta ,0)=\zeta \), where \(K(x)=\frac{1}{2\pi }\frac{x}{ \left\vert x\right\vert ^{2}}\). He proves the existence of a \(C^{\infty }\) initial data \(f_{0}(x,v)\) leading to a \(C^{\infty }\) solution \(f(x,v, t)\) to the above Vlasov-Poisson system such that the some trajectory \((X(\zeta _{0},t),V(\zeta _{0},t))\) is not analytic in time at \(t=0\).

MSC:

76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
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