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Transport-diffusion equations with irregular data and applications to stability estimates for second-order Hamilton-Jacobi PDEs

Gianmarco Giovannardi, Alessandro Goffi

TL;DR

The paper addresses quantitative uniqueness and stability for Fokker-Planck and transport-diffusion equations with irregular drift by introducing two structural conditions on the divergence of the velocity field: (i) a mixed Lebesgue condition $\mathrm{div}(b)\in L^r_t(L^q_x)$ with $\tfrac{n}{2q}+\tfrac{1}{r}\leq 1$, and (ii) a one-sided time-singular bound on $[\mathrm{div}(b(t))]^-$ allowing $\dfrac{c}{t}$ blow-up. It derives explicit, data-dependent $L^p$ and $L^{\infty}$ stability and uniqueness results through energy estimates and duality with the adjoint Fokker-Planck equation, and then applies these to obtain continuous dependence estimates for viscous Hamilton-Jacobi equations without relying on viscosity-solution theory. The work extends the framework to unbounded domains and Neumann boundary conditions and provides higher-order and $L^1$ stability results under Aronson-Serrin-type interpolations, offering new insight into the stability of nonlinear HJ equations with rough data. The duality-based approach yields explicit constants and broad applicability to degenerate or nonlocal operators, potentially impacting homogenization and Mean Field Game analyses.

Abstract

This paper studies quantitative uniqueness properties in $L^p$ spaces for Fokker-Planck and transport-diffusion equations under two new assumptions on their velocity field $b=b(x,t)$. We first prove $L^p$-stability estimates for advection-diffusion PDEs when $\mathrm{div}(b)\in L^r_t(L^q_x)$ with $r\in[2,\infty]$ and $q\in[n/2,\infty)$ satisfying the compatibility condition $n/(2q)+1/r\leq 1$. We then prove a stability result in $L^\infty$ for solutions of viscous transport equations when $\mathrm{div}(b(t))$ fails to be integrable in time. We apply these properties to obtain new continuous dependence estimates for viscous Hamilton-Jacobi equations via integral methods. One of the main novelties in this latter setting is that the constants of the estimates are all explicit with respect to the data of the problem. These imply new uniqueness properties for diffusive Hamilton-Jacobi equations without relying on the theory of viscosity solutions.

Transport-diffusion equations with irregular data and applications to stability estimates for second-order Hamilton-Jacobi PDEs

TL;DR

The paper addresses quantitative uniqueness and stability for Fokker-Planck and transport-diffusion equations with irregular drift by introducing two structural conditions on the divergence of the velocity field: (i) a mixed Lebesgue condition with , and (ii) a one-sided time-singular bound on allowing blow-up. It derives explicit, data-dependent and stability and uniqueness results through energy estimates and duality with the adjoint Fokker-Planck equation, and then applies these to obtain continuous dependence estimates for viscous Hamilton-Jacobi equations without relying on viscosity-solution theory. The work extends the framework to unbounded domains and Neumann boundary conditions and provides higher-order and stability results under Aronson-Serrin-type interpolations, offering new insight into the stability of nonlinear HJ equations with rough data. The duality-based approach yields explicit constants and broad applicability to degenerate or nonlocal operators, potentially impacting homogenization and Mean Field Game analyses.

Abstract

This paper studies quantitative uniqueness properties in spaces for Fokker-Planck and transport-diffusion equations under two new assumptions on their velocity field . We first prove -stability estimates for advection-diffusion PDEs when with and satisfying the compatibility condition . We then prove a stability result in for solutions of viscous transport equations when fails to be integrable in time. We apply these properties to obtain new continuous dependence estimates for viscous Hamilton-Jacobi equations via integral methods. One of the main novelties in this latter setting is that the constants of the estimates are all explicit with respect to the data of the problem. These imply new uniqueness properties for diffusive Hamilton-Jacobi equations without relying on the theory of viscosity solutions.
Paper Structure (8 sections, 12 theorems, 103 equations)

This paper contains 8 sections, 12 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The case of vector fields with divergence in mixed Lebesgue spaces
  3. $L^p$ a priori estimates and uniqueness
  4. Transport-diffusion equations with non-integrable one-side conditions on $\mathrm{div}(b)$
  5. Applications to uniqueness and continuous dependence estimates of Hamilton-Jacobi equations
  6. Semiconcavity-type condition (i)
  7. The "weak" Aronson-Serrin interpolated condition (ii)
  8. The Aronson-Serrin interpolated condition (iii)

Key Result

Theorem 2.1

Assume that Then every solution $\rho$ in the parabolic class $\mathcal{H}_2^1(Q_T)$ of eq:fp satisfies the estimate where $C_1$ depends on $n,q,r,\|\mathrm{div}(b)\|_{L^r_t(L^q_x)}$. Morevoer, we have where $C_2$ depends on $n,q,r,\|\mathrm{div}(b)\|_{L^r_t(L^q_x)}, T$.

Theorems & Definitions (36)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • ...and 26 more