Martingale Solutions Research Articles

Stochastic Cahn–Hilliard and conserved Allen–Cahn equations with logarithmic potential and conservative noise*

Abstract We investigate the Cahn–Hilliard and the conserved Allen–Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure that the order parameter takes its values in the physical range and, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. For the Cahn–Hilliard equation, existence and uniqueness of probabilistically-strong solutions is shown up to the three-dimensional case. For the conserved Allen–Cahn equation, under a restriction on the noise magnitude, existence of martingale solutions is proved even in dimension three, while existence and uniqueness of probabilistically-strong solutions holds in dimension one and two. The analysis is carried out by studying the Cahn–Hilliard/conserved Allen–Cahn equations jointly, that is a linear combination of both the equations, which has an independent interest.

Dean–Kawasaki equation with initial condition in the space of positive distributions

We show that the Dean–Kawasaki equation does not admit non-trivial solutions in the space of tempered measures. More specifically, we consider martingale solutions taking values, and with initial conditions, in the subspace of measures admitting infinite mass and satisfying some integrability conditions. Following work by the first author, Lehmann and von Renesse (Electron Commun Probab 24:3916340, 2019), we show that the equation only admits solutions if the initial measure is a discrete measure. Our result extends the previously mentioned works by allowing measures with infinite mass.

Global dissipative martingale solutions to the variational wave equation with stochastic forcing

Global dissipative martingale solutions to the variational wave equation with stochastic forcing

Well-posedness for the stochastic Landau–Lifshitz–Gilbert equation with helicity driven by jump noise

We consider the stochastic Landau–Lifshitz–Gilbert equation driven by pure jump noise. We assume non-zero contribution from the helicity term to the total energy. Using finite dimensional approximation followed by a generalization of the Jakubowski’s version of the Skorohod Theorem for non-metric spaces, we show that the considered problem admits a weak martingale solution. Restricting the problem to dimension 1, we show that the obtained solution is pathwise unique, thereby concluding the existence of a strong solution.

Martingale solutions and asymptotic behaviors for a stochastic cross-diffusion three-species food chain model with prey-taxis.

The stochastic food chain model is an important model within the field of ecological research. Since existing models are difficult to describe the influence of cross-diffusion and random factors on the evolution of species populations, this work is concerned with a stochastic cross-diffusion three-species food chain model with prey-taxis, in which the direction of predators' movement is opposite to the gradient of prey, i.e., a higher density of prey. The existence and uniqueness of martingale solutions are established in a Hilbert space by using the stochastic Galerkin approximation method, the tightness criterion, Jakubowski's generalization of the Skorokhod theorem, and the Vitali convergence theorem. Furthermore, asymptotic behaviors around the steady states of the stochastic cross-diffusion three-species food chain model in the time mean sense are investigated. Finally, numerical simulations are carried out to illustrate the results of our analysis.

Martingale solution of the stochastic Camassa–Holm equation with pure jump noise

We study the stochastic Camassa–Holm equation with pure jump noise. We establish the existence of the global martingale solution by the regularization method, the tightness criterion, the generalization of the Skorokhod theorem for nonmetric spaces and the stochastic renormalized formulations.

Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.

Well-posedness for the stochastic Landau–Lifshitz–Bloch equation with helicity

We consider the stochastic Landau–Lifshitz–Bloch equation with helicity term driven by a real-valued Wiener process. We show the existence of a weak martingale solution, followed by pathwise uniqueness of the obtained solution, culminating into the existence of a strong solution using the theory of Yamada and Watanabe.

Martingale Solutions in Stochastic Fluid–Structure Interaction

We consider a viscous incompressible fluid interacting with a linearly elastic shell of Koiter type which is located at some part of the boundary. Recently models with stochastic perturbation in the shell equation have been proposed in the literature but only analysed in simplified cases. We investigate the full model with transport noise, where (a part of) the boundary of the fluid domain is randomly moving in time. We prove the existence of a weak martingale solution to the underlying system.

Martingale solutions to the stochastic thin-film equation in two dimensions

Martingale solutions to the stochastic thin-film equation in two dimensions

Global martingale weak solutions for the three-dimensional stochastic chemotaxis-Navier-Stokes system with Lévy processes

This paper considers the three-dimensional stochastic chemotaxis-Navier-Stokes (SCNS) system subjected to a Lévy-type random external force in a bounded domain. Until recently, the existed results concerning global solvability of SCNS system mainly concentrated on the case of two spatial dimensions, little is known about the SCNS system in dimension three. We prove in the present work that the initial-boundary value problem for the three-dimensional SCNS system has at least one global martingale solution under proper assumptions, which is weak in both the analytical and stochastic sense. A new stochastic analogue of entropy-energy inequality and an uniform boundedness estimate are derived, which enable us to construct global-in-time approximate solutions from a suitably regularized SCNS system via the Contraction Mapping Principle. The proof of the existence of martingale solution is based on a stochastic compactness method and an elaborate identification of the limits procedure, where the Jakubowski-Skorokhod Theorem is applied to deal with the phase spaces equipped with weak topology.

Global existence of dissipative solutions to the Camassa–Holm equation with transport noise

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.

Martingale solutions and invariant measures for the stochastic strongly damped wave equation with critical nonlinearity

Martingale solutions and invariant measures for the stochastic strongly damped wave equation with critical nonlinearity

Invariant measures for a stochastic nonlinear and damped 2D Schrödinger equation

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin’s method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well.

Weak solution for stochastic Degasperis-Procesi equation

This article is concerned with the existence of solution to the stochastic Degasperis-Procesi equation on R with an infinite dimensional multiplicative noise and integrable initial data.Writing the equation as a system composed of a stochastic nonlinear conservation law and an elliptic equation, we are able to develop a method based on the conjugation of kinetic theory with stochastic compactness arguments. More precisely, we apply the stochastic Jakubowski-Skorokhod representation theorem to show the existence of a weak kinetic martingale solution. In this framework, the solution is a stochastic process with sample paths in Lebesgue spaces, which are compatible with peakons and wave breaking physical phenomenon.

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.

Stochastic wave equation with Hölder noise coefficient: Well-posedness and small mass limit

We construct unique martingale solutions to the damped stochastic wave equationμ∂2u∂t2(t,x)=Δu(t,x)−∂u∂t(t,x)+b(t,x,u(t,x))+σ(t,x,u(t,x))dWtdt, where Δ is the Laplacian on [0,1] with Dirichlet boundary condition, W is space-time white noise, σ is 34+ϵ -Hölder continuous in u and uniformly non-degenerate, and b has linear growth. The same construction holds for the stochastic wave equation without damping term. More generally, the construction holds for SPDEs defined on separable Hilbert spaces with a densely defined operator A, and the assumed Hölder regularity on the noise coefficient depends on the eigenvalues of A in a quantitative way. We further show the validity of the Smoluchowski-Kramers approximation: assume b is Hölder continuous in u, then as μ tends to 0 the solution to the damped stochastic wave equation converges in distribution, on the space of continuous paths, to the solution of the corresponding stochastic heat equation. The latter result is new even in the case of additive noise.

A short remark on inviscid limit of the stochastic Navier–Stokes equations

In this article, we study the inviscid limit of the stochastic incompressible Navier–Stokes equations in three-dimensional space. We prove that a subsequence of weak martingale solutions of the stochastic incompressible Navier–Stokes equations converges strongly to a weak martingale solution of the stochastic incompressible Euler equations in the periodic domain under the well-accepted hypothesis, namely Kolmogorov hypothesis (K41).

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

A dynamic capillarity equation with stochastic forcing on manifolds: A singular limit problem

We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold ( M , g ) (M,g) : d ( u ε , δ − δ Δ u ε , δ ) + d i v f ε ( x , u ε , δ ) d t = ε Δ u ε , δ d t + Φ ( x , u ε , δ ) d W t , \begin{equation*} d \left (u_{\varepsilon ,\delta } -\delta \Delta u_{\varepsilon ,\delta }\right ) +div\mathfrak {f}_{\varepsilon }(\mathbf {x}, u_{\varepsilon ,\delta })\, dt =\varepsilon \Delta u_{\varepsilon ,\delta }\, dt + \Phi (\mathbf {x}, u_{\varepsilon ,\delta })\, dW_t, \end{equation*} where f ε \mathfrak {f}_{\varepsilon } is a sequence of smooth vector fields converging in L p ( M × R ) L^p(M\times \mathbb {R}) ( p > 2 p>2 ) as ε ↓ 0 \varepsilon \downarrow 0 towards a vector field f ∈ L p ( M ; C 1 ( R ) ) \mathfrak {f}\in L^p(M;C^1(\mathbb {R})) , and W t W_t is a Wiener process defined on a filtered probability space. First, for fixed values of ε \varepsilon and δ \delta , we establish the existence and uniqueness of weak solutions to the Cauchy problem for the above-stated equation. Assuming that f \mathfrak {f} is non-degenerate and that ε \varepsilon and δ \delta tend to zero with δ / ε 2 \delta /\varepsilon ^2 bounded, we show that there exists a subsequence of solutions that strongly converges in L ω , t , x 1 L^1_{\omega ,t,\mathbf {x}} to a martingale solution of the following stochastic conservation law with discontinuous flux: d u + d i v f ( x , u ) d t = Φ ( u ) d W t . \begin{equation*} d u +div\mathfrak {f}(\mathbf {x}, u)\,dt =\Phi (u)\, dW_t. \end{equation*} The proofs make use of Galerkin approximations, kinetic formulations as well as H H -measures and new velocity averaging results for stochastic continuity equations. The analysis relies on the use of a.s. representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.