The kinetic Fokker-Planck equation with weak confinement force (Q2292673)

The purpose of this paper is to determine a rate of convergence on the trend to equilibrium for the kinetic Fokker-Planck (KFP) equation. The equation being studied is \(\partial_t f=\mathcal{L}f := -v\cdot \nabla_xf+\nabla_x V(x)\cdot \nabla_v f + \Delta_v f +\operatorname{div}_v(vf),\) where \(f=f(t,x,v)\) is the phase space distribution function with \(x \in \mathbb{R}^d\), \(v \in \mathbb{R}^d\), and \(t \ge 0\). The initial data is \(f(0,\cdot) = f_0(x,v)\). The confinement potential \(V\) is of the form \[ V = \langle x \rangle^{\gamma},\ \gamma \in (0,1), \] and \(\langle x \rangle = 1+|x|^2\). The function \[ G = Z^{-1}e^-{W},\ W = \frac{|v|^2}{2} + V(x), \] is a positive, normalized steady state for equation (1) given an appropriate normalizing constant \(Z\). The present paper deals with the problem of weak confinement for which \(\gamma \in (0,1)\) in the expression for \(V\). The case of strong confinement corresponds to \(\gamma \ge 1\). To prove the main theorem of the paper, weighted \(L^p\) spaces are defined as \(L^p(m) = \{f:fm \in L^p \}\) for the weight function \(m\). The main result of the paper is given as a theorem (Theorem 1.1) which states: \begin{itemize} \item[(1)] For initial data \(f_0 \in L^p(G^{-(\frac{p-1}{p}+\epsilon)})\), \(p \in [1, \infty)\), \(\epsilon > 0\) small, the solution to (1.1) converges to the equilibrium state at a rate \(\leq De^{-Ct^b}\) for any \(b \in (0,\frac{\gamma}{2-\gamma})\) and constants \(C>0\), \(D>0\). \item[(2)] If \(f_0 \in L^1(m)\), \(m=H^k\), \(H=|x|^2+|v|^2+1\), \(k \ge 1\), then the solution to (1) converges to the steady state with rate \(\le D(1+t)^{-\frac{k}{1-\frac{\gamma}{2}}}\). \end{itemize} The proof of Theorem (1.1) involves defining various subspaces of \(L^2\) using \(G\) in different ways as a weight function. These subspaces act as interpolation spaces in \(L^2\). Convergence estimates on the solution \(f\) to (1) are obtained in the interpolation spaces. These estimates are then generalized to decay estimates on a wider class of \(L^p\) spaces. This is done by obtaining \(L^p\) estimates on the semigroup associated with the operator \(\mathcal{L}\) in terms of \(L^2\) estimates. A~point the author makes is that a Poincaré inequality that holds for strong confinement, \(\gamma \ge 1\), does not hold in the case of weak confinement, \(0 < \gamma < 1\). As a part of the proof, the author provides a~weaker version of the Poincaré inequality that can apply in the case of weak confinement. The author points out that there already exist in the literature convergence proofs providing a rate of convergence for the case of strong confinement. Also, some papers have previously been published proving convergence in the case of weak confinement. The new results in the present paper are to prove for the case of weak confinement an exponential (rather than geometric) rate of convergence for the KFP equation using deterministic (rather than probabilistic) methods. Also, the analysis in the paper broadens the class of initial data for which a rate of convergence is obtained.