arXiv subjects

Nathaël Alibaud

Publications and source records attributed to Nathaël Alibaud.

10 recordsLinked to original sources

arXiv2025

Nonlocal degenerate parabolic hyperbolic equations on bounded domains

We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\text{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion operator $\mathcal{L}$ can be any symmetric L{é}vy operator (e.g. fractional Laplacians) and $b$ is nondecreasing and allowed to have degenerate regions ($b'=0$). We propose an entropy solution formulation for the problem and show uniqueness of bounded entropy solutions under general assumptions. Existence of solutions is shown in a separate paper. The uniqueness proof is based on the Kružkov doubling of variables technique and incorporates several a priori results derived from our entropy formulation: an $L^\infty$-bound, an energy estimate, strong initial trace, weak boundary traces, and a \textit{nonlocal} boundary condition. Our work can be seen as both extending nonlocal theories from the whole space to domains and local theories on domains to the nonlocal case. Unlike local theories our formulation does not assume energy estimates. They are now a consequence of the formulation, but as opposed to previous nonlocal theories, they play an essential role in our arguments.

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arXiv2024

Optimal stability results and nonlinear duality for $L^\infty$ entropy and $L^1$ viscosity solutions

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality: \begin{equation}\int |S_t u_0-S_t v_0| φ_0 \mathrm{d}x\leq \int |u_0-v_0| G_t φ_0 \mathrm{d}x, \quad \forall φ_0 \geq 0, \forall u_0, \forall v_0, \qquad(\star)\end{equation} where $S_t$ is the entropy solution semigroup of the anisotropic degenerate parabolic equation \begin{equation*} \partial_t u+\mathrm{div} F(u) = \mathrm{div} (A(u) D u),\end{equation*} and where we look for the smallest semigroup $G_t$ satisfying ($\star$). This amounts to finding an optimal weighted $L^1$ contraction estimate for $S_t$. Our main result is that $G_t$ is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation\begin{equation*} \partial_t φ= \mathrm{sup}_ξ\{F'(ξ) \cdot D φ+\mathrm{tr}(A(ξ) D^2φ)\}.\end{equation*} Since weighted $L^1$ contraction results are mainly used for possibly nonintegrable $L^\infty$ solutions $u$, the natural spaces behind this duality are $L^\infty$ for $S_t$ and $L^1$ for $G_t$. We therefore develop a corresponding $L^1$ theory for viscosity solutions $φ$. But $L^1$ itself is too large for well-posedness, and we rigorously identify the weakest $L^1$ type Banach setting where we can have it -- a subspace of $L^1$ called $L^\infty_{\mathrm{int}}$. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].

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arXiv2020

The Liouville theorem and linear operators satisfying the maximum principle

A result by Courrège says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ where $$ \mathcal{L}^{σ,b}[u](x)=\text{tr}(σσ^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^μ[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} μ(z). $$ This class of operators coincides with the infinitesimal generators of Lévy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$. The proofs combine arguments from PDE and group theories. They are simple and short.

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arXiv2019

(Bounded) Traveling Combustion Fronts With Degenerate Kinetics

We consider the propagation of a flame front in a solid periodic medium. The model is governed by a free boundary system in which the front's velocity depends on the temperature via a kinetic rate which may degenerate. We show the existence of travelling wave solutions which are bounded and global. Previous results by the same authors (cf. Alibaud, Namah, On the propagation of a periodic flame front by an Arrhenius kinetic,{\it Interfaces Free Bound.,} 2017) were obtained for essentially positively lower bounded kinetics or eventually which have some very weak degeneracy. Here we consider general degenerate kinetics, including in particular those of Arrhenius type which are commonly used in physics.

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arXiv2019

Nonlocal Dissipation Measure and L 1 Kinetic Theory For Fractional Conservation Laws

We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L 1 initial data, general self-adjoint pure jump L{é}vy operators, and locally Lipschitz nonlinearities of porous medium kind possibly strongly degenerate. The cornerstone of the formulation and the uniqueness proof is an adequate explicit representation of the nonlocal dissipation measure. This approach is inspired from the second order theory unlike the cutting technique previously introduced for bounded entropy solutions. The latter technique no longer seems to fit our setting. This is moreover the first time that the more standard and sharper tools of the second order theory are faithfully adapted to fractional conservation laws.

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arXiv2018

Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of $\mathbb{R}^d$

We investigate the characterization of generators $\mathcal{L}$ of Lévy processes satisfying the Liouville theorem: Bounded functions $u$ solving $\mathcal{L}[u]=0$ are constant. These operators are degenerate elliptic of the form $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ for some local part $\mathcal{L}^{σ,b}[u]=\text{tr}(σσ^{\texttt{T}} D^2u)+b \cdot Du$ and nonlocal part $$ \mathcal{L}^μ[u](x)=\int \big(u(x+z)-u(x)-z \cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \, \text{d} μ(z), $$ where $μ\geq 0$ is a so-called Lévy measure possibly unbounded for small $z$. In this paper, we focus on the pure nonlocal case $σ=0$ and $b=0$, where we assume in addition that $μ$ is symmetric which corresponds to self-adjoint pure jump Lévy operators $\mathcal{L}=\mathcal{L}^μ$. The case of general Lévy operators $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ will be considered in the forthcoming paper \cite{AlDTEnJa18}. In our setting, we show that $\mathcal{L}^μ[u]=0$ if and only if $u$ is periodic wrt the subgroup generated by the support of $μ$. Therefore, the Liouville property holds if and only if this subgroup is dense, and in space dimension $d=1$ there is an equivalent condition in terms of irrational numbers. In dimension $d \geq 1$, we have a clearer view of the operators \textit{not} satisfying the Liouville theorem whose general form is precisely identified. The proofs are based on arguments of propagation of maximum.

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arXiv2016

On the propagation of a periodic flame front by an arrhenius kinetic

We consider the propagation of a flame front in a solid medium with a periodic structure. The model is governed by a free boundary system for the pair" temperature-front. "The front's normal velocity depends on the temperature via a (degenerate) Arrhenius kinetic. It also depends on the front's mean curvature. We show the existence of travelling wave solutions for the full system and consider their homogenization as the period tends to zero. We analyze the curvature effects on the homogenization and obtain a continuum of limiting waves parametrized by the limiting ratio "curvature coefficient/period." This analysis provides valuable information on the heterogeneous propagation as well.

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arXiv2013

Continuous dependence estimates for nonlinear fractional convection-diffusion equations

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.

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arXiv2010

Asymptotic properties of entropy solutions to fractal Burgers equation

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{α/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for alpha in (1,2), the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for alpha \leq 1. If alpha=1, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case alpha \in (0,1), we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions.

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arXiv2007

A non-monotone conservation law for dune morphodynamics

We investigate a non-local non linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see \cite{Fow01, Fow02}. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well-posedness for initial data in $L^2$ and give explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results.

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