[Paper Review] Stability of Entropy Solutions for Levy Mixed Hyperbolic-Parabolic Equations

Kenneth H. Karlsen, Süleyman Ulusoy|ArXiv.org|Feb 3, 2009

Nonlinear Partial Differential Equations38 references18 citations

TL;DR

This paper establishes the uniqueness and continuous dependence of entropy solutions for a class of Lévy mixed hyperbolic-parabolic equations with non-local (fractional) diffusion driven by pure jump Lévy processes. Using a front-tracking and doubling variable technique, the authors prove an $L^1$ contraction property and stability under perturbations of flux, diffusion, and jump measures, extending classical results to non-local settings.

ABSTRACT

We analyze entropy solutions for a class of Levy mixed hyperbolicparabolic equations containing a non-local (or fractional) diffusion operator originating from a pure jump Levy process. For these solutions we establish uniqueness (L1 contraction property) and continuous dependence results.

Motivation & Objective

• To establish the uniqueness of entropy solutions for a class of degenerate parabolic-hyperbolic equations with non-local (fractional) diffusion arising from pure jump Lévy processes.
• To prove continuous dependence of solutions on initial data, flux functions, diffusion coefficients, and jump measures.
• To extend the $L^1$ contraction property (uniqueness) to equations with non-local operators beyond the classical local diffusion case.
• To analyze the stability of solutions under perturbations of the non-local generator $\mathcal{L}$, particularly in the context of singular integral representations of Lévy measures.
• To provide a rigorous framework for entropy solutions in the presence of mixed hyperbolic-parabolic behavior with non-local diffusion, generalizing existing results to non-local settings.

Proposed method

• Uses a doubling of variables technique combined with front-tracking approximations to handle discontinuities in solutions.
• Employs a singular integral representation of the non-local operator $\mathcal{L}$ as $\mathcal{L}[u](t,x) = \int_{\mathbb{R}^d \setminus \{0\}} \left[ u(t,x+z) - u(t,x) - z \cdot \nabla u \mathbf{1}_{|z|<1} \right] \pi(dz)$.
• Applies the Lévy-Khintchine formula to characterize the non-local generator $\mathcal{L}$ as a pure jump process with zero drift and zero diffusion, relying on a Lévy measure $\pi$ satisfying integrability conditions.
• Introduces a regularized version of the $L^1$-norm using mollifiers $\delta_\mu$ and time-regularization $\theta_\nu$ to handle the non-smoothness of entropy solutions.
• Estimates the difference between two entropy solutions via integral terms involving flux, diffusion, and non-local parts, decomposing the total variation into $I_{\text{conv}}, I_{\text{diff}}, I_{\text{fdiff}}$.
• Uses integration by parts and BV regularity ($u \in L^\infty(BV)$) to control the non-local terms, particularly $I_{\text{fdiff}_1}$ and $I_{\text{fdiff}_2}$, by bounding them via $\|m - \tilde{m}\|_{L^1}$ and $\|\sigma^a - \sigma^b\|_{L^\infty}$.

Experimental results

Research questions

• RQ1 Does the $L^1$ contraction property hold for entropy solutions of hyperbolic-parabolic equations with non-local (fractional) diffusion?
• RQ2 How does the solution depend continuously on the initial data, flux function, diffusion coefficient, and jump measure in the non-local operator?
• RQ3 Can the classical uniqueness and stability results for local parabolic equations be extended to equations with pure jump Lévy processes?
• RQ4 What is the role of the Lévy measure $\pi$ in the stability of entropy solutions, especially when $\pi$ is singular or non-Gaussian?
• RQ5 How can the doubling variable method be adapted to handle non-local operators with non-smooth kernels and singular integrals?

Key findings

• The paper proves the $L^1$ contraction property for entropy solutions, establishing uniqueness in the class $L^\infty(0,T;L^1 \cap L^\infty)(\mathbb{R}^d)$.
• Continuous dependence on initial data is established with the bound $\|u(t,\cdot) - v(t,\cdot)\|_{L^1} \leq \|u_0 - v_0\|_{L^1} + C t \|f - g\|_{\text{Lip}(I)} + C t \|\sigma^a - \sigma^b\|_{L^\infty} + C t \|m - \tilde{m}\|_{L^1}$.
• The limit $\lim_{\alpha \to 0} \lim_{\nu \to 0} |I_{\text{fdiff}_1}| \leq \frac{C}{\mu} \tau \int_{|z|<1} |z|^2 |m(z) - \tilde{m}(z)| \, dz$ is derived, showing control of the non-local part via the $L^1$-norm of the jump measure difference.
• For the heavy-tailed part, $\lim_{\nu \to 0} \lim_{\alpha \to 0} |I_{\text{fdiff}_2}| \leq C \tau \int_{|z| \geq 1} |z| |m(z) - \tilde{m}(z)| \, dz$ is obtained, ensuring stability under large jumps.
• The convergence of the regularized terms is controlled: $\lim_{\nu \to 0} R_t = 0$, $\limsup_{\alpha \to 0} |R_x| \leq C \mu$, and $\lim_{\alpha \to 0} L = \|u(\tau,\cdot) - v(\tau,\cdot)\|_{L^1} - \|u_0 - v_0\|_{L^1}$.
• Optimization over $\mu$ yields the final continuous dependence estimate (2.8), confirming stability under perturbations of all system components.