Nonlinear Stochastic Partial Differential Equations Research Articles

Wong–Zakai approximation for a stochastic 2D Cahn–Hilliard–Navier–Stokes model

AbstractIn this paper, we demonstrate the Wong–Zakai approximation results for two dimensional stochastic Cahn–Hilliard–Navier–Stokes model. The model consists of a Navier–Stokes system coupled with convective Cahn–Hilliard equations. It describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids under the influence of multiplicative noise. Our main result describes the support of the distribution of solutions. As in [2], both inclusions are proved by means of a general Wong–Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert‐valued Brownian motion and some adapted finite‐dimensional approximation of this process. Note that the coupling between the Navier–Stokes system and the Cahn–Hilliard equations makes the analysis more involved.

Non-intrusive reduced-order model for time-dependent stochastic partial differential equations utilizing dynamic mode decomposition and polynomial chaos expansion

In this study, we present a novel non-intrusive reduced-order model (ROM) for solving time-dependent stochastic partial differential equations (SPDEs). Utilizing proper orthogonal decomposition (POD), we extract spatial modes from high-fidelity solutions. A dynamic mode decomposition (DMD) method is then applied to vertically stacked matrices of projection coefficients for future prediction of coefficient fields. Polynomial chaos expansion (PCE) is employed to construct a mapping from random parameter inputs to the DMD-predicted coefficient field. These lead to the POD–DMD–PCE method. The innovation lies in vertically stacking projection coefficients, ensuring time-dimensional consistency in the coefficient matrix for DMD and facilitating parameter integration for PCE analysis. This method combines the model reduction of POD with the time extrapolation strengths of DMD, effectively recovering field solutions both within and beyond the training time interval. The efficiency and time extrapolation capabilities of the proposed method are validated through various nonlinear SPDEs. These include a reaction–diffusion equation with 19 parameters, a two-dimensional heat equation with two parameters, and a one-dimensional Burgers equation with three parameters.

Stochastic electromechanical bidomain model *

We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.

Stochastic evolution equations with Wick-analytic nonlinearities

We study nonlinear stochastic partial differential equations with Wick-analytic type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fisher–KPP equations, stochastic Allen–Cahn, stochastic Newell–Whitehead–Segel, and stochastic Fujita–Gelfand equations. By implementing the theory of C 0 − semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove the existence and uniqueness of solutions for this class of stochastic partial differential equations.

Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations

Recently, Hairer et al. (2015) showed that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong L p L^p -sense with respect to the initial value for every p ∈ ( 0 , ∞ ] p\in (0,\infty ] . In this article we provide conditions on the coefficient functions of the SDE and on p ∈ ( 0 , ∞ ] p \in (0,\infty ] that are sufficient for local Lipschitz continuity in the strong L p L^p -sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear stochastic ordinary and stochastic partial differential equations in the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.

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.

Simulating a strongly nonlinear backward stochastic partial differential equation via efficient approximation and machine learning

<abstract><p>We have studied a strongly nonlinear backward stochastic partial differential equation (B-SPDE) through an approximation method and with machine learning (ML)-based Monte Carlo simulation. This equation is well-known and was previously derived from studies in finance. However, how to analyze and solve this equation has remained a problem for quite a long time. The main difficulty is due to the singularity of the B-SPDE since it is a strongly nonlinear one. Therefore, by introducing new truncation operators and integrating the machine learning technique into the platform of a convolutional neural network (CNN), we have developed an effective approximation method with a Monte Carlo simulation algorithm to tackle the well-known open problem. In doing so, the existence and uniqueness of a 2-tuple adapted strong solution to an approximation B-SPDE were proved. Meanwhile, the convergence of a newly designed simulation algorithm was established. Simulation examples and an application in finance were also provided.</p></abstract>

A generalized finite element θ-scheme for backward stochastic partial differential equations and its error estimates

In this paper, we study numerical methods for solving a class of nonlinear backward stochastic partial differential equations. By utilizing finite element methods in space and θ-scheme in time, the proposed scheme forms a generalized spatio-temporal full discrete scheme, which can be solved in parallel. We rigorously prove the boundedness and error estimates, and obtain the optimal convergence rates in both time (first order/second order) and space (k + 1, k in L2 and H1, respectively). Numerical results are finally provided to demonstrate the effectiveness of the proposed scheme and validate the theoretical analyses.

Effects of Wiener process on analytical wave solutions for (3+1) dimensional nonlinear Schrödinger equation using modified extended mapping method

In this study, we investigate the soliton solutions for stochastic (3+1) dimensional nonlinear Schrödinger equation (NLSE). Our Study mainly depends on applying the modified extended mapping method to construct various and novel stochastic solutions for the proposed model. These solutions including (dark, bright, and singular) solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, and Jacobi elliptic function (JEF) solutions are raised. For a variety of nonlinear stochastic partial differential equations, this method offers a practical and effective method for determining exact stochastic solutions. The stochastic noise effects forcing the wave to behave differently from deterministic cases prognosticated. To illustrate the physical characteristics of the established stochastic solutions, some selected solutions are presented graphically. We used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.

Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type

Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type

Long-time behavior of stochastic Hamilton-Jacobi equations

The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a regularization by noise phenomenon for the mean curvature flow with homogeneous noise which establishes that the inclusion of noise speeds up the decay of solutions, and (ii) the long-time convergence of solutions to spatially inhomogeneous stochastic Hamilton-Jacobi equations. A number of motivating examples about nonlinear stochastic partial differential equations are presented in the appendix.

The exponential stability for locally monotone stochastic partial differential equations

This paper mainly studies the exponential stability for a class of nonlinear stochastic partial differential equations with locally monotone coefficients. The generalized variational framework is adopted to analyze the mean square exponential stability and pathwise exponential stability. Additionally, the use of multiplicative Itô noise with sufficient strength to enhance the stability of solution to stochastic partial differential equation is explored when the mean square exponential stability does not hold.

Oblique projection for scalable rank-adaptive reduced-order modelling of nonlinear stochastic partial differential equations with time-dependent bases

Time-dependent basis reduced-order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive and highly parallelizable. These favourable properties are achieved via oblique projections that require evaluating the MDE at a small number of rows and columns. The columns and rows are selected using the discrete empirical interpolation method (DEIM), which yields near-optimal matrix low-rank approximations. We show that the proposed algorithm is equivalent to a CUR matrix decomposition. Numerical results demonstrate the accuracy, efficiency and robustness of the new method for a diverse set of problems.

Talagrand’s transportation inequality for SPDEs with locally monotone drifts

The purpose of this paper is twofold. Firstly, we prove transportation inequalities T2(C) on the space of continuous paths with respect to the uniform metric for the law of the solution to a class of non-linear monotone stochastic partial differential equations (SPDEs) driven by the Wiener noise. Furthermore, we also establish the T1(C) property for such SPDEs but with merely locally monotone coefficients, including the stochastic Burgers type equation and stochastic 2-D Navier–Stokes equation.

Computing non-equilibrium trajectories by a deep learning approach

Predicting the occurrence of rare and extreme events in complex systems is a fundamental problem in non-equilibrium physics. These events can have huge impacts on human societies. New approaches have emerged in recent years, which better estimate tail distributions. They often use large deviation concepts without the need to perform heavy direct ensemble simulations. In particular, a well-known approach is to derive a minimum action principle and find its minimizers.The analysis of rare reactive events in non-equilibrium systems without detailed balance is notoriously difficult both theoretically and computationally. They can often be described in the limit of small noise by the Freidlin-Wentzell action. Here, we propose a method which minimizes the geometric action instead using neural networks: it is called deep gMAM. It relies on a natural and simple machine-learning formulation of the classical gMAM approach when the Lagrangian is known. We provide a detailed description of the method as well as many examples: from bimodal switches in nonlinear stochastic (partial) differential equations, including a codimension-1 nucleation in an Allen-Cahn-Ginzburg-Landau 5-D stochastic partial differential equation, to quasi-potential estimates and extreme events in Burgers turbulence.

A selfdual variational principle with minimal hypothesis and applications to stationary, dynamic and stochastic equations

We revisit the self-dual variational approach to the resolution of non-linear stationary, parabolic and stochastic partial differential equations involving a monotone non-linearity, by relaxing the hypothesis under which it is applicable, in two directions: The prohibitive coercivity condition which was problematic for applications to evolution equations, be them deterministic or stochastic, and the restrictive “regularity” property that required the domain of the non-linear operator under study to be large. The new principle is applied to give more direct proofs for the existence of weak solutions for incompressible Navier-Stokes equations in their stationary, dynamic or stochastic versions.

Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities

This paper aims at studying a generalized Camassa–Holm equation under random perturbation. We establish a local well-posedness result in the sense of Hadamard, i.e., existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s$$\\end{document} with s>3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s>3/2$$\\end{document} for x∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\in \\mathbb R$$\\end{document}. The analysis on continuous dependence on initial data for nonlinear stochastic partial differential equations has gained less attention in the literature so far. In this work, we first show that the solution map is continuous. Then we introduce a notion of stability of exiting time. We provide an example showing that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data. Finally, we analyze the regularization effect of nonlinear noise in preventing blow-up. Precisely, we demonstrate that global existence holds true almost surely provided that the noise is strong enough.

A mathematical modeling and numerical study for stochastic Fisher–SI model driven by space uniform white noise

In this study, we propose an approximate solution based on two‐dimensional shifted Legendre polynomials to solve nonlinear stochastic partial differential equations with variable coefficients. For this purpose, we have considered a Fisher‐Kolmogorov‐Petrovsky‐Piskunov (Fisher–KPP) equation with space uniform white noise for the same. New stochastic operational matrix of integration based on shifted Legendre polynomials is generated. This operational matrix reduces the problem under study into solving a system of algebraic equations. The convergence analysis is discussed, and the error bound in norm is derived and obtained as . The theoretical analysis confirms that as the degree of the approximation polynomial is increased, the solution is on par with the exact solution. The numerical examples confirm the accuracy and the applicability of the proposed method. A comparative study is carried out with explicit 1.5 order Runge–Kutta method. The time complexity of the proposed technique is also studied.

Incompressible Euler equations with stochastic forcing: A geometric approach

We consider a stochastic version of Euler equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden (1970). For the Euler equations on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic non-linear partial differential equations.

Solving a nonlinear fractional SPDE with spatially inhomogeneous white noise

In this paper, we study the following nonlinear fractional stochastic partial differential equation where denotes the Markovian generator of a stable-like Feller process with variable order and is a measurable function. The forcing noise denoted by is a spatially inhomogeneous white noise. Under some assumptions on the catalytic measure of the inhomogeneous Brownian sheet , we study the moment bounds for the solution. As a byproduct, we prove that the solution is weakly full intermittent based on the moment estimates of the solution. We also study the Hölder regularity of the solution with respect to the temporal and spatial variables, respectively.