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On uniqueness of solutions to nonlinear Fokker-Planck-Kolmogorov equations. (English) Zbl 1336.35334
The authors study the uniqueness of the Cauchy problem for a nonlinear Fokker-Planck-Kolmogorov equation of the form \(\partial_t \mu = \partial_{x_i}\partial_{x_j}(a^{i,j}(\mu,x,t)\mu) - \partial_{x_i}(b^i(\mu,x,t),\mu)\) with \(\mu|_{t=0}= \mu_0\). A solution is a finite Borel \(\mu(dx\, dt) = \mu_t(dx)\) given by a family of probability measures \((\mu_t)_{t\in[0,T]}\) on \(\mathbb{R}^d\). Hence, the equations is to be understood in the sense of distributions. The diffusion matrix \(A=(a^{ij})\) is assumed to be symmetric and non-negative definite. The main goal is to establish sufficient conditions for uniqueness allowing for nonsmooth and unbounded coefficients.
One way to establish well-posedness of the above equation is to use the connection to the martingale problem, where standard assumptions on the coefficients are a global Lipschitz bound. This restricts the nonlinear dependence of the coefficients \(A\) and \(b\) on \(\mu\) to be a convolution with an at most polynomial growing kernel. Therefore the authors aim to establish uniqueness for non-Lipschitz coefficients with faster growing nonlinearities.
The proof is based on the Holmgren method: First, the adjoint problem \(\partial_s f + L_\mu f=0\) with \(f|_{s=t} = \psi\) and \(\psi \in C_0^\infty(\mathbb{R}^d)\) is solved in the class of sufficiently smooth functions, where \(L_\mu = a^{i,j} \partial_{x_i} \partial_{x_j} + b^i \partial_{x_i}\) is the formal adjoint. Then, for two solutions \(\mu,\sigma\) with \(\mu|_{t=0}= \mu_0\) and \(\sigma|_{t=0}=\sigma_0\) one can formally deduce an a priori estimate of the form \(W_1(\mu_t,\sigma_t) \leq e^{Ct} W_1(\mu_0,\sigma_0)\) with for instance \(W_1\) the Kantorovich metric.
The main difficulty in this approach is solving the adjoint problem with nonregular and unbounded coefficients. Depending on the choice of the assumptions on the coefficients, there will be three steps necessary: Find a suitable approximation of the coefficients. Solve the adjoint problem. Find a suitable metric to which the Holmgren method can be applied. These steps are implemented for three classes of coefficients.
The first case assumes the diffusion matrix \(A\) to be elliptic and independent of the solution. Under the assumption of the existence of a suitable Lyapunov function as well as additional regularity and growth assumptions on \(A\) and \(b\) the existence is established via Holmgren methods wrt. a weighted \(L^1\)-norm.
The second case considers degenerate but still independent diffusion matrix \(A\), the extreme case being \(A\equiv 0\) Since the solution \(\mu\) has no reason to be absolute continuous wrt. Lebesgue, a family of metrics on the space of probability measures is introduced. For a continuous convex function \(W\) on \(\mathbb{R}^d\) with \(W\geq 1\) and measures \(\mu,\sigma\) such that \(|x| \sqrt{W(x)}\in L^1(\mu + \sigma)\) the metric is defined by \(w_W(\mu,\sigma)=\sup\{ \int f d(\mu-\sigma) : f\in C_0^\infty(\mathbb{R}^d), |\nabla f(x)| \leq \sqrt{W(x)}\}\). This metrics generalize the Kantorovich \(1\)-metric, which is obtained by the choice \(W\equiv 1\). A particular choice leading to a metric on measures with \(p\)th moment is \(W(x) = (1+|x|^{p-1})^2\), where the corresponding metric is denoted by \(w_p\). This one can then be compared with the usual Kantorovich \(p\)-metric. The Holmgren method can then be implemented under suitable assumptions on the vectorfield \(b\).
The last case deals with a non-degenerate but dependent diffusion matrix \(A\). The ellipticity allows to still estimated the second derivatives of the solution to the adjoint problem in terms of first derivatives, which then connects back to the metric \(w_W\). Hence, the essential assumption, besides regularity and growth control of \(A\) and \(b\) in space, is a modulus of continuity of the dependence of the coefficients on the solution wrt. the metric \(w_W\). This allows to implement Holmgren method wrt. the metric \(w_W\) also in this case. The authors give in the last section examples of non-uniqueness in the case of a degenerate and dependent diffusion matrix \(A\). This also illustrates the necessity of certain assumptions made before.

MSC:
35Q84 Fokker-Planck equations
35K55 Nonlinear parabolic equations
35Q83 Vlasov equations
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B45 A priori estimates in context of PDEs
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