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Paul Razafimandimby

Publications and source records attributed to Paul Razafimandimby.

11 recordsLinked to original sources
arXiv2025

Existence of weak solution of 3D ferrohydrodynamic equations with transport noise: Bloch-Torrey regularisation

In this article, we consider a stochastic ferrohydrodynamic system which describes the Bloch-Torrey regularization of the motion of an electrically conducting ferrofluids driven by transport noise filling a 3D bounded domain with a smooth boundary. We prove the global existence of a probabilistic weak solution of the stochastic system by making use of the combination of Galerkin method and compactness method. The system under study is basically a coupling of the Navier-Stokes equations with internal rotation, the Maxwell equations and the ferromagnetization equations. Thus, our result can be seen as a modest generalization of the existing results on the global existence of weak solutions of stochastic Magnetohydrodynamic (MHD), Navier-Stokes and Bloch type equations on 3D bounded domain.

math.AP↗
arXiv2025

Weak solutions of Navier-Stokes Equation with purely discontinuous Lévy Noise

In this paper we prove the existence of weak martingale solutions to the stochastic Navier-Stokes Equations driven by pure jump Lévy processes. Our proof consists of two parts. In the first one, mostly classical, we recall a priori estimates, from the paper by the third named author, for solutions to suitable constructed Galerkin approximations and we use the Jakubowski-Skorokhod Theorem to find a sequence of processes on a new probability space convergent point-wise to a limit process. In the second one, we show that the limit process is a weak martingale solution to the SNSEs by using an approach of Kallianpur and Xiong. The core of this method consists of a proof that a certain natural process on the new probability space is a purely discontinuous martingale and then to use a suitable representation theorem. In this way we propose a method of proving solutions to stochastic PDEs which is different from the method used recently by the first and fourth named author in their joint paper with E.\ Hausenblas, see \cite{Brz+Haus+Raza_2018_reaction_diffusion}.

math.PR↗
arXiv2020

Strong solution to stochastic penalised nematic liquid crystals model driven by multiplicative Gaussian noise

In this paper, we prove the existence of a unique maximal local strong solutions to a stochastic system for both 2D and 3D penalised nematic liquid crystals driven by multiplicative Gaussian noise. In the 2D case, we show that this solution is global.As a by-product of our investigation, but of independent interest, we present a general method based on fixed point arguments to establish the existence and uniqueness of a maximal local solution of an abstract stochastic evolution equations with coefficients satisfying local Lipschitz condition involving the norms of two different Banach spaces.

math.AP↗
arXiv2018

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.

math.PR↗
arXiv2017

Existence of a density of the 2Dim Stochastic Navier Stokes Equation driven by Levy processes or fractional Brownian motion

In this article we are interested in the regularity properties of the probability measure induced by the solution process of the Lévy noise or a fractional Brownian motion driven Navier Stokes Equation on the two dimensional torus $\mathbb{T}$. We mainly investigate under which conditions on the characteristic measure of the Lévy process or the Hurst parameter of the fractal Brownian motion the law of the projection of $u(t)$ onto any finite dimensional $F\subset L^2(\mathbb{T})$ is absolutely continuous with respect to the Lebesgue measure on $F$.

math.PR↗
arXiv2016

Some results on the penalised nematic liquid crystals driven by multiplicative noise

In this paper we prove several results related to the existence and uniqueness of solution to coupled highly nonlinear stochastic partial differential equations (PDEs). These equations are motivated by the dynamics of nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of global weak solution (in sense of both stochastic analysis and PDEs). We show the pathwise uniqueness of the solution in 2D domain. Secondly, we establish the existence and uniqueness of local maximal solution which is strong in sense of both PDEs and stochastic analysis. In the 2D case, we show that this solution is global. In contrast to several works in the deterministic setting we replace the Ginzburg-Landau function ${1}_{\lvert \mathbf{n}\rvert \le 1}(\lvert \mathbf{n}\rvert^2-1)\mathbf{n}$ by a general polynomial $f(\mathbf{n})$ and we give sufficient conditions on the polynomial $f$ for the aforementioned results to hold. As a by-product of our investigation we present a general method based on fixed point argument to establish the existence and uniqueness of a local maximal solution of an abstract stochastic evolution equations with coefficients satisfying local Lipschitz condition involving the norms of two different Banach spaces. This general method can be used to treat several stochastic hydrodynamical models such as Navier-Stokes, Magnetohydrodynamic (MHD) equations, and the $α$-models of Navier-Stokes equations and their MHD counterparts.

math.PR↗
arXiv2015

Analytic properties of Markov semigroup generated by Stochastic Differential Equations driven by Lévy processes

We consider the stochastic differential equations of the form \begin{equation*} \begin{cases} dX^ x(t) = σ(X(t-)) dL(t) \\ X^ x(0)=x,\quad x\in\mathbb{R}^ d, \end{cases} \end{equation*} where $σ:\mathbb{R}^ d\to \mathbb{R}^ d$ is Lipschitz continuous and $L=\{L(t):t\ge 0\}$ is a Lévy process. Under this condition on $σ$ it is well known that the above problem has a unique solution $X$. Let $(\mathcal{P}_{t})_{t\ge0}$ be the Markovian semigroup associated to $X$ defined by $( \mathcal{P}_t f) (x) := \mathbb{E} [ f(X^ x(t))]$, $t\ge 0$, $x\in \mathbb{R}^d$, $f\in \mathcal{B}_b(\mathbb{R}^d)$. Let $B$ be a pseudo--differential operator characterized by its symbol $q$. Fix $ρ\in\mathbb{R}$. In this article we investigate under which conditions on $σ$, $L$ and $q$ there exist two constants $γ>0$ and $C>0$ such that $$ \lvert B \mathcal{P}_t u \rvert_{H^ρ_2} \le C \, t^{-γ} \,\lvert u \rvert_{H^ρ_2}, \quad \forall u \in {H^ρ_2}(\mathbb{R}^d ),\, t>0. $$

math.PR↗
arXiv2014

Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type

In this paper we prove the existence and uniqueness of maximal strong (in PDE sense) solution to several stochastic hydrodynamical systems on unbounded and bounded domains of $\mathbb{R}^n$, $n=2,3$. This maximal solution turns out to be a global one in the case of 2D stochastic hydrodynamical systems. Our framework is general in the sense that it allows us to solve the Navier-Stokes equations, MHD equations, Magnetic Bénard problems, Boussinesq model of the Bénard convection, Shell models of turbulence and the Leray-$α$ model with jump type perturbation. Our goal is achieved by proving general results about the existence of maximal and global solution to an abstract stochastic partial differential equations with locally Lipschitz continuous coefficients. The method of the proofs are based on some truncation and fixed point methods.

math.PR↗
arXiv2014

On the rate of convergence of the 2-D stochastic Leray-$α$ model to the 2-D stochastic Navier-Stokes equations with multiplicative noise

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-$α$ model to the solution of the 2-D stochastic Navier-Stokes equations. We are mainly interested in the rate of convergence, as $α$ tends to 0, of the error function which is the difference between the two solutions in an appropriate topology. We show that when properly localized the error function converges in mean square as $α\to 0$ and the convergence is of order $O(α)$. We also prove that the error converges in probability to zero with order at most $O(α)$.

math.PR↗
arXiv2011

Convergence of a sequence of solutions of the stochastic two-dimensional equations of second grade fluids

We study the limit of the stochastic model for two dimensional second grade fluids subjected to the periodic boundary conditions as the stress modulus tends to zero. We show that under suitable conditions on the data the whole sequence of strong probabilistic solutions $(u^α)$ of the stochastic second grade fluid converges to the unique strong probabilistic solution of the stochastic Navier-Stokes equations.

math.AP↗