arXiv subjects
Publications and source records attributed to Julien Vovelle.
We prove a quantitative averaging lemma for spatially dependent vector fields. Our proof is based on an iteration of the regularizing operator and some elementary considerations about the local inversion theorem.
We investigate the Smoluchowski-Kramers approximation for the one-dimensional periodic variational wave equation with state-dependent damping and additive noise. We show that weak ``dissipative'' solutions converge to solutions of a stochastic quasilinear parabolic equation.
We consider the variational wave equation in one-dimensional space with stochastic forcing by an additive noise. Blow-up of local smooth solutions is established, and global existence is proved in the class of weak martingale solutions.
We prove the existence of global-in-time regular solutions to a system of stochastic quadratic reaction-diffusion equations. Global-in-time existence is based on a $L^\infty$-estimate obtained by an approach {à} la De Giorgi, as in [GoudonVasseur10]. The adaptation of this technique to the stochastic case requires in its final step an $L^2\ln(L^2)$-bound, furnished by an estimate by duality on the entropy inequality, as in [DesvillettesFellnerPierreVovelle07]. In our stochastic context, and similarly to [DebusscheRoselloVovelle2021], we need to solve a backward SPDE to exploit the duality technique
We derive the hydrodynamic limit of a kinetic equation with a stochastic, short range perturbation of the velocity operator. Under some mixing hypotheses on the stochastic perturbation, we establish a diffusion-approximation result: the limit we obtain is a parabolic stochastic partial differential equation on the macroscopic parameter, the density here.
We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier-Stokes perturbative regime. On the torus, we establish local-in-time, for any time, Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of the characteristics trajectories due to the presence of the force. This uniform theory is sufficiently robust to derive the incompressible Navier-Stokes-Fourier system with an external force from the Boltzmann equation. Neither smallness, nor time-decaying assumption is required for the external force, nor a gradient form, and we deal with general hard potential and cut-off Boltzmann kernels. As a by-product the latest general theories for unit Knudsen number when the force is sufficiently small and decays in time are recovered.
We study a kinetic toy model for a spray of particles immersed in an ambient fluid, subject to some additional random forcing given by a mixing, space-dependent Markov process. Using the perturbed test function method, we derive the hydrodynamic limit of the kinetic system. The law of the limiting density satisfies a stochastic conservation equation in Stratonovich form, whose drift and diffusion coefficients are completely determined by the law of the stationary process associated with the Markovian perturbation.
We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modelled by a linear operator (Fokker-Planck or Linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.
We develop a general framework for the analysis of approximations to stochastic scalar conservation laws. Our aim is to prove, under minimal consistency properties and bounds, that such approximations are converging to the solution to a stochastic scalar conservation law. The weak probabilistic convergence mode is convergence in law, the most natural in this context. We use also a kinetic formulation and martingale methods. Our result is applied to the convergence of the Finite Volume Method in the companion paper [15].
We study the kinetic Fokker-Planck equation perturbed by a stochastic Vlasov force term. When the noise intensity is not too large, we solve the Cauchy Problem in a class of well-localized (in velocity) functions. We also show that, when the noise intensity is sufficiently small, the system with prescribed mass admits a unique invariant measure which is exponentially mixing. The proof uses hypocoercive decay estimates and hypoelliptic gains of regularity. At last we also exhibit an explicit example showing that some restriction on the noise intensity is indeed required.
We prove the convergence of the explicit-in-time Finite Volume method with monotone fluxes for the approximation of scalar first-order conservation laws with multiplicative, compactly supported noise.
In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized Itô formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.
We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity ($L^p$-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow.
We study the stochastically forced system of isentropic Euler equations of gas dynamics with a $γ$-law for the pressure. We show the existence of martingale weak entropy solutions; we also discuss the existence and characterization of invariant measures in the concluding section.
The aim of this paper is the rigorous derivation of a stochastic non-linear diffusion equation from a radiative transfer equation perturbed with a random noise. The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The Hilbert expansion has to be done up to order 3 to overcome some diffculties caused by the random noise.
We study the stochastic diffusive limit of a kinetic radiative transfer equation, which is non-linear, involving a small parameter and perturbed by a smooth random term. Under an appropriate scaling for the small parameter, using a generalization of the perturbed test-functions method, we show the convergence in law to a stochastic non-linear fluid limit.
We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.
Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour of periodic scalar first-order conservation laws with stochastic forcing in any space dimension. For sub-cubic fluxes, we show the existence of an invariant measure. Moreover for sub-quadratic fluxes we show uniqueness and ergodicity of the invariant measure. Also, since this invariant measure is supported by Lp for some p small, we are led to generalize to the stochastic case the theory of L1 solutions developed by Chen and Perthame in 2003.