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Well-posedness results for superlinear Fokker-Planck equations

Stefano Buccheri, Fernando Farroni, Gabriella Zecca

TL;DR

This work addresses the well-posedness of nonlinear parabolic Fokker-Planck equations with space-dependent, superlinear drift in bounded domains, aiming for solutions in $C([0,T);L^1)$. It develops a nonvariational framework using energy-type estimates, truncation, and compactness to obtain global or local-in-time existence depending on the drift exponent $\theta$, and it establishes $L^1$ contraction and $L^p$ regularity results. For small $\theta$, global existence is obtained; for larger $\theta$, the estimates may blow up in finite time, reflecting noncoercivity of the drift. The paper also treats the nonzero right-hand side through a fixed-point approach, showing small-data global existence via Schauder's theorem, and provides a robust set of existence, regularity, and uniqueness results that extend prior work beyond variational settings.

Abstract

In this manuscript we deal with a class of nonlinear Fokker-Planck equations with the following structure \[ \partial_t u - ÷\big(M\nabla u+ E h(u)\big)=0, \] with $M$ a bounded elliptic matrix, $E$ a vector field in a suitable Lebesgue space, and $h(u)$ featuring a superlinear growth for $u$ large. We provide existence results of $C([0,T),L^1)$ distributional solutions to initial-boundary value problems related to the equation above together with some qualitative properties of solutions.

Well-posedness results for superlinear Fokker-Planck equations

TL;DR

This work addresses the well-posedness of nonlinear parabolic Fokker-Planck equations with space-dependent, superlinear drift in bounded domains, aiming for solutions in . It develops a nonvariational framework using energy-type estimates, truncation, and compactness to obtain global or local-in-time existence depending on the drift exponent , and it establishes contraction and regularity results. For small , global existence is obtained; for larger , the estimates may blow up in finite time, reflecting noncoercivity of the drift. The paper also treats the nonzero right-hand side through a fixed-point approach, showing small-data global existence via Schauder's theorem, and provides a robust set of existence, regularity, and uniqueness results that extend prior work beyond variational settings.

Abstract

In this manuscript we deal with a class of nonlinear Fokker-Planck equations with the following structure with a bounded elliptic matrix, a vector field in a suitable Lebesgue space, and featuring a superlinear growth for large. We provide existence results of distributional solutions to initial-boundary value problems related to the equation above together with some qualitative properties of solutions.
Paper Structure (6 sections, 15 theorems, 186 equations)

This paper contains 6 sections, 15 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Notation and functional framework
  3. Preliminary results
  4. Proof of Theorems \ref{['thetasmalf=0']} and \ref{['thetabigf=0']}
  5. Proof of Theorem \ref{['exist2']}
  6. Proof of Theorem \ref{['thmSte']}

Key Result

Theorem 1.1

Let us assume mcon, thetasmall, and take $r\in(N,\infty]$ such that Then, for any given $T>0$, $u_0\in L^\mu(\Omega)$ with $\mu\ge1$, and $|E|\in L^{\infty}((0,\infty);L^r(\Omega))$, there exists a unique weak solution $u\in C([0,T];L^1(\Omega))\cap L^2_{loc}((0,T); W^{1,2}_0(\Omega))$ to introintro such that with $C_1=C_1(\alpha, N, \theta, \mu, r, T, E, \|u_0\|_{L^{1}(\Omega)})$. Moreover for

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Proposition 3.1
  • ...and 23 more