Radjesvarane Alexandre
We give simple proofs of hypoelliptic estimates for some models of kinetic equations with a fractional order diffusion part. The proofs are based on energy estimates together with F. Bouchut and B. Perthame previous ideas.
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We consider the passive scalar equations subject to shear flow advection and fractional dissipation. The enhanced dissipation estimates are derived. For classical passive scalar equation ($γ=1$), our result agrees with the sharp one obtained in \cite{Wei18}
In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.
In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.
Averaging lemmas and hypoellipticity
本文使用交换子和基本解方法，为纯动力学输运方程建立平均化引理和亚椭圆估计，并在一般非齐次通量条件下推导出正则性性质，应用于平流-扩散方程及双曲守恒律的动力学形式。
The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir \cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To remedy this issue, we shall modify the usual $H^k$ norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time (c.f. \cite{Herau}), usually referred as Hérau's method, can be employ…
We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family $f_λ(x,v)\in L^p$ satisfying some appropriate transport relation $$v\cdot\nabla_x f_λ= (1-Δ_x)^\fracβ{2}(1-Δ_v)^\fracα{2}g_λ$$ may be inferred solely from its compactness in $v$. This method is introduced as an alternative to the lack of known suitable averaging lemmas in $L^1$ when the right-hand side of the transport equation has very low regularity, due to an e…
The voltage-conductance equation determines the probability distribution of a stochastic process describing a fluctuation-driven neuronal network arising in the visual cortex. Its structure and degeneracy share many common features with the kinetic Fokker-Planck equation, which has attracted much attention recently. We prove an L $\infty$ bound on the steady state solution and the long term convergence of the evolution problem towards this stationary state. Despite the hypoellipticity property, the difficulty is to treat the boundary conditions that change type along the boundary. This leads u…
We establish the existence and symmetry of all minimizers of a constrained variational problem involving the fractional gradient. This problem is closely connected to some fractional kinetic equations.
We prove regularization properties in short time for inhomogeneous kinetic equations whose collision kernel behaves like a fractional power of the Laplacian in velocity. We treat a fractional Kolmogorov equation and the linearized Boltzmann equation without cutoff (for hard potentials).
We study the kinetic Fokker-Planck equation on the whole space with a confining potential. We show quantitative rates of exponential convergence to equilibrium in a well chosen Wasserstein-1 distance. We use the Wasserstein-1 version of Harris's theorem introduced by Hairer and Mattingly. We make use of similarities between hypocoercivity and hypoellipticity in order to use Malliavin calculus to see hypocoercivity for this equation on the level of the SDE.
We develop a new generalized form of the fractional kinetic equation involving a generalized k-Bessel function. The generalized $k$-Mittag-leffler function $E^{γ,q}_{k,α,β}(.)$ is discussed in terms of the solution of the fractional kinetic equation in the present paper. The results obtained here are quite general in nature and capable of yielding known and as well new results.
This paper is devoted to hydrodynamic limits of linear kinetic equations when the thermodynamical equilibrium is described by a heavy-tail distribution function rather than a Maxwellian distribution. We show that the long time/small mean free path behavior of the solution of the kinetic equation is described by a fractional diffusion equation. The method that we introduce is somewhat reminiscent of the so-called "moments method" which plays an important role in kinetic theory.
A version of fractional diffusion on bounded domains, subject to 'homogeneous Dirichlet boundary conditions' is derived from a kinetic transport model with homogeneous inflow boundary conditions. For nonconvex domains, the result differs from standard formulations. It can be interpreted as the forward Kolmogorow equation of a stochastic process with jumps along straight lines, remaining inside the domain.
In this paper, we develop an operator splitting scheme for the fractional kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a fractional diffusion phase and a kinetic transport phase. The first phase is solved exactly using the convolution operator while the second one is solved approximately using a variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. We prove the convergence of the scheme to a weak solution to FKFPE. As a by-product of our analysis, we also establish a variational formulation …
Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation
该研究引入带时间权重的动力学极大$L^2$正则性，用分数阶各向异性Sobolev空间刻画（分数阶）Kolmogorov方程解的正则性，并证明齐次方程解构成半流且具有瞬时正则化性质。
A fractional diffusion equation with advection term is rigorously derived from a kinetic transport model with a linear turning operator, featuring a fat-tailed equilibrium distribution and a small directional bias due to a given vector field. The analysis is based on bounds derived by relative entropy inequalities and on two recently developed approaches for the macroscopic limit: a Fourier-Laplace transform method for spatially homogeneous data and the so called moment method, based on a modified test function.
Kinetic maximal $L^p_μ(L^p)$-regularity for the fractional Kolmogorov equation with variable density
本文研究带可变密度的非局部分数阶 Kolmogorov 方程，证明其动力学极大 $L^p_\mu$-正则性，并用于建立拟线性分数阶动力学 PDE 强解的短时间存在性。
This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.
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 fi…
We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows H{ö}lder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. The theory of hypoellipticity of H{ö}rmander shows, under general "bracket" conditions, the regularity of solutions to partial differential equations combining first and second order derivative operators when ellipticity fails in some directions. We discuss recent exten…
Kinetic-transport equations that take into account the intra-cellular pathways are now considered as the correct description of bacterial chemotaxis by run and tumble. Recent mathematical studies have shown their interest and their relations to more standard models. Macroscopic equations of Keller-Segel type have been derived using parabolic scaling. Due to the randomness of receptor methylation or intra-cellular chemical reactions, noise occurs in the signaling pathways and affects the tumbling rate. Then, comes the question to understand the role of an internal noise on the behavior of the f…
In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0 < $α$ < 1. We turn the free interface problem into a scalar free boundary problem coupled with an integral equation. The main intermediary step is to reduce the scalar problem to the study of a non-C 1 vector field in dimension 2. The latter is treated by qualitative topo-logical methods based on the Poincar{é}-Bendixson Theorem. The phase portrait is determined and the existence of a stable manifold at the…
Fractional Hypocoercivity
本文研究无约束动理学方程在胖尾局部平衡碰撞算子下的大时间行为，发展$L^2$-亚矫顽性方法，证明其衰减率与反常扩散极限及分数阶Nash型不等式相容。
In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws driven by a multiplicative noise. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theor…
We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.
Intrinsic Hölder spaces for fractional kinetic operators
该研究引入与伽利略几何结构相关的各向异性Hölder空间，证明内蕴类泰勒公式，推广了李群上纯微分算子的相关结果，为分数阶非局部动力学算子正则性研究提供工具。
Well-posedness of the Fractional Fokker-Planck Equation
本文利用Schauder型估计方法，建立了分数阶Fokker-Planck方程的关键适定性，并展示了该方法对非截断Boltzmann方程和Landau方程的潜在适用性。
In this paper we use an individual-based model and its associated kinetic equation to study the generation of long jumps in the motion of E. coli. These models relate the run-and-tumble process to the intracellular reaction where the intrinsic noise plays a central role. Compared with the previous work in [13] in which the parametric assumptions are mainly for mathematical convenience and not well-suited for either numerical simulation or comparison with experimental results, our current paper make use of biologically meaningful pathways and tumbling kernels. Moreover, using the individual-bas…
Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $σ$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $α$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $σ\to+\infty$ and $1-α\sim C/σ$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a…
This paper is devoted to the anomalous diffusion limit of kinetic equations with a fractional Fokker-Planck collision operator in a spatially bounded domain. We consider two boundary conditions at the kinetic scale: absorption and specular reflection. In the absorption case, we show that the long time/small mean free path asymptotic dynamics are described by a fractional diffusion equation with homogeneous Dirichlet-type boundary conditions set on the whole complement of the spatial domain. On the other hand, specular reflections will give rise to a new operator which we call specular diffusio…
We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the {\it unique solution} of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to pro…
We investigate the fractional diffusion approximation of a kinetic equation in the upper-half plane with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we derive a fractional diffusion equation with a nonlocal Neumann boundary condition for the density of particles. Interestingly, this asymptotic equation is different from the one derived by L. Cesbron in [7] in the case of specular reflection conditions at the boundary and does not seem to have receive a lot of attention previously.
Uniqueness of traveling fronts in premixed flames with stepwise ignition-temperature kinetics and fractional reaction order
本文研究具有点火温度动力学与分数反应阶数的预混火焰反应-扩散系统，证明行波锋面解在平移意义下唯一，并通过渐近与数值分析推测传播速度随各参数递减。
Macroscopic boundary conditions for a fractional diffusion equation in chemotaxis
本文针对趋化性中分数阶扩散方程，利用边界层方法和摄动理论推导对称反射条件下内部解与边界层解的一阶近似，揭示分数阶动力学与边界相互作用的机制。
Slow-fast system in Rosales-Majda combustion model with fractional order kinetics
本文研究了罗萨尔斯-马尔达燃烧模型中慢-快系统的行波解及粘性冲击波的分支图特性。
We consider the limit of a linear kinetic equation, with reflection-transmission-absorption at an interface, with a degenerate scattering kernel. The equation arise from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation, with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, re…
In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmann-type equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavy-tailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision frequency for small velocities. This work aims to an extension in the sense that a linear BGK-type equation conserving not only mass, but also momentum and energy, for both mentioned regimes of equilibrium d…
This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result…
Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations
本文提出带时间权重的动力学极大 $L^p$-正则性理论，证明其适用于分数阶及变系数 Kolmogorov 方程，并用于建立一类拟线性动力学方程解的局部存在性。
We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to the deterministic case and as far as we work in $L^2$, the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right hand side belongs to $L^2$), or regularity is better when the right hand side contains deriva…
The $\mathcal{M}$-Operator and Uniqueness of Nonlinear Kinetic Equations
本文提出一种$\mathcal{M}$-算子方法，通过避免对解的导数有界性的假设，建立了Landau型非线性动力学方程连续或有界解的唯一性，适用于低正则性解。
The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $(-Δ)^α$. Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $α\ge 1$ in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations…
The Harnack inequality fails for nonlocal kinetic equations
本研究证明非局部动力学方程的Harnack不等式不成立，以分数阶Kolmogorov方程为反例，指出这是纯粹非局部现象，补充了非局部亚椭圆方程理论的相关结论。
The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Levy distribution. This article clarifies the form of biologically relevant model equations: We derive Patlak-Keller-Segel-like equations involving nonlocal, fractional Laplacians from a microscopic model for cell movement. Starting from a power-law distribution of run times, we derive a kinetic equation in which the collision term takes into account the long range behaviour of the individuals. A fractional chemotactic equation is obtained in a bi…
On the spectral problem and fractional diffusion limit for Fokker-Planck with(-out) drift and for a general heavy tail equilibrium
本文研究一般重尾平衡态下动力学 Fokker–Planck 算子的谱问题，证明唯一特征对存在，并据此建立带或不带漂移方程的分数阶扩散极限。
Kinetic SDEs and kinetic Fokker-Planck equations with critical Morrey-type drifts, and local blow-ups
该研究针对受奇异Morrey型力场的动力学Fokker-Planck方程建立Sobolev理论，通过分数表示与动力学Adams型估计构造布朗驱动动力学SDE的弱解，得到相关估计与传播子并建立弱解唯一性准则。
We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-absorption condition at the interface. Both the coefficient describing the probability of absorption and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to t…
Bound states and heat kernels for fractional-type Schrödinger operators with singular potentials
本文研究带分数阶 Hardy 势的非局部 Schrödinger 算子，建立负特征值特征函数的逐点估计、热核有限时间上界及其与基态下界的联系。
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