Nonlinear Stochastic Evolution Equations Research Articles

Rate-independent stochastic evolution equations: Parametrized solutions

By extending to the stochastic setting the classical vanishing viscosity approach we prove the existence of suitably weak solutions of a class of nonlinear stochastic evolution equation of rate-independent type. Approximate solutions are obtained via viscous regularization. Upon properly rescaling time, these approximations converge to a parametrized martingale solution of the problem in rescaled time, where the rescaled noise is given by a general square-integrable cylindrical martingale with absolutely continuous quadratic variation. In absence of jumps, these are strong-in-time martingale solutions of the problem in the original, not rescaled time.

On the Impact of Noise on Hyperbolic-Type Traveling Wave Solutions of Some Stochastic Evolution Equations

In this paper, traveling wave solutions of some nonlinear stochastic evolution equations emerging in miscellaneous fields such as modeling of flame propagation, magneto-acoustic waves in plasma and small-amplitude water waves with surface tension are investigated. By means of Galilean transformation and tanh method, we obtain some exact solutions such as kink wave solution, solitary wave solution and periodic solution. To illustrate the impact of noise on the solutions, we assigned different noise functions for the external noise. The results showed that the waveform deforms as the noise intensity increases.

Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence* *The second author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).

In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an L p -setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers’ equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.

Stability of delay evolution equations with fading stochastic perturbations

ABSTRACT Stability of nonlinear delay evolution equation with stochastic perturbations is considered. It is shown that if the level of stochastic perturbations fades on the infinity then an exponentially stable deterministic system remains to be exponentially stable (in mean square). This idea has already been checked for ordinary linear stochastic differential equations and linear stochastic difference equations. Here a similar statement is proven for a nonlinear stochastic evolution equation with quickly enough fading stochastic perturbations. More exactly: if the level of the stochastic perturbation is given by a continuous and square integrable function, then the zero solution of the considered exponentially stable deterministic system remains exponentially mean square stable independently on the maximum magnitude of the stochastic perturbations. Applications of the obtained results to stochastic reaction-diffusion equations and stochastic 2D Navier-Stokes model are shown. Consideration of other types of fading stochastic perturbations (for instance, if stochastic perturbations fade on the infinity not so quickly) is an open problem.

Averaging principle for impulsive stochastic partial differential equations

This paper focuses on systems of stochastic partial differential equations with impulse effects. We establish an averaging principle such that the solution to the complex original nonlinear impulsive stochastic evolution equations can be approximated by that to the more simplified averaged stochastic evolution equations without impulses. By adopting stochastic analysis theory, semigroup approach and inequality technique, sufficient conditions are formulated and the mean square convergence is proved. This ensures that we can concentrate on the averaged system instead of the original system, thus providing a solution for reduction of complexity.

Random Attractors for 2D Stochastic Hydrodynamical-Type Systems

We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the 3D Leray 𝛼-model. We prove the existence of random attractors for the associated continuous random dynamical systems. We also establish the upper semicontinuity of the random attractors as the parameter tends to zero.

Doubly nonlinear stochastic evolution equations

Nonlinear diffusion problems featuring stochastic effects may be described by stochastic partial differential equations of the form [Formula: see text] We present an existence theory for such equations under general monotonicity assumptions on the nonlinearities. In particular, [Formula: see text], [Formula: see text], and [Formula: see text] are allowed to be multivalued, as required by the modelization of solid–liquid phase transitions. In this regard, the equation corresponds to a nonlinear-diffusion version of the classical two-phase Stefan problem with stochastic perturbation. The existence of martingale solutions is proved via regularization and passage-to-the-limit. The identification of the limit is obtained by a lower-semicontinuity argument based on a suitably generalized Itô’s formula. Under some more restrictive assumptions on the nonlinearities, existence and uniqueness of strong solutions follows. Besides the relation above, the theory covers equations with nonlocal terms as well as systems.

Square-mean piecewise almost automorphic mild solutions to a class of impulsive stochastic evolution equations

In this paper, we introduce the concept of square-mean piecewise almost automorphic function. By using the theory of semigroups of operators and the contraction mapping principle, the existence of square-mean piecewise almost automorphic mild solutions for linear and nonlinear impulsive stochastic evolution equations is investigated. In addition, the exponential stability of square-mean piecewise almost automorphic mild solutions for nonlinear impulsive stochastic evolution equations is obtained by the generalized Gronwall–Bellman inequality. Finally, we provide an illustrative example to justify the results.

Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the d-dimensional torus with singular p-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative Itô-formula.

Consistent nonlinear deterministic and stochastic wave evolution equations from deep water to the breaking region

Modelling the evolution of the wave field in coastal waters is a complicated task, partly due to triad nonlinear wave interactions, which are one of the dominant mechanisms in this area. Stochastic formulations already implemented into large-scale operational wave models, whilst very efficient, are one-dimensional in nature and fail to account for the majority of the physical properties of the wave field evolution. This paper presents new two-dimensional (2-D) formulations for the triad interactions source term. A quasi-two-dimensional deterministic mild slope equation is improved by including dissipation and first-order spatial derivatives in the nonlinear part of equation, significantly enhancing the accuracy in the breaking zone. The newly defined deterministic model is used to derive an updated stochastic model consistent from deep waters to the breaking region. It is localized following the approach derived in Vrecica & Toledo (J. Fluid Mech., vol. 794, 2016, pp. 310–342), to which several improvements are also presented. The model is compared to measurements of breaking and non-breaking spectral evolution, showing good agreement in both cases. Finally, the model is used to analyse several interesting 2-D properties of the shoaling wave field including the evolution of directionally spread seas.

A class of stochastic hyperbolic evolution equations via weighted pseudo almost periodic coefficients and optimal controls

SummaryThis paper introduces the concept of the piecewise weighted pseudo periodicity for stochastic processes. In addition, it establishes a new composition theorem for weighted pseudo almost periodic functions under the non‐Lipschitz conditions. Using this composition theorem, evolution family together with a fixed‐point theorem for condensing maps, we investigate the existence of p‐mean piecewise weighted pseudo almost periodic mild solutions and optimal mild solutions for a class of impulsive stochastic hyperbolic evolution equations in Hilbert spaces. Then, the existence conditions of optimal pairs of systems governed by the nonlinear impulsive stochastic evolution equations are presented. Finally, an example is provided to illustrate the obtained theory.

Existence and stability of μ-pseudo almost automorphic solutions for stochastic evolution equations

We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.

Fractional Measure-dependent Nonlinear Second-order Stochastic Evolution Equations with Poisson Jumps

Abstract This paper focuses on a nonlinear second-order stochastic evolution equations driven by a fractional Brownian motion (fBm) with Poisson jumps and which is dependent upon a family of probability measures. The global existence of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory.

Nonlinear stochastic evolution equations of second order with damping

Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with an internal approximation. Uniqueness is proved as well.

Consistent nonlinear stochastic evolution equations for deep to shallow water wave shoaling

Nonlinear interactions between sea waves and the sea bottom are a major mechanism for energy transfer between the different wave frequencies in the near-shore region. Nevertheless, it is difficult to account for this phenomenon in stochastic wave forecasting models due to its mathematical complexity, which mostly consists of computing either the bispectral evolution or non-local shoaling coefficients. In this work, quasi-two-dimensional stochastic energy evolution equations are derived for dispersive water waves up to quadratic nonlinearity. The bispectral evolution equations are formulated using stochastic closure. They are solved analytically and substituted into the energy evolution equations to construct a stochastic model with non-local shoaling coefficients, which includes nonlinear dissipative effects and slow time evolution. The nonlinear shoaling mechanism is investigated and shown to present two different behaviour types. The first consists of a rapidly oscillating behaviour transferring energy back and forth between wave harmonics in deep water. Owing to the contribution of bottom components for closing the class III Bragg resonance conditions, this behaviour includes mean energy transfer when waves reach intermediate water depths. The second behaviour relates to one-dimensional shoaling effects in shallow water depths. In contrast to the behaviour in intermediate water depths, it is shown that the nonlinear shoaling coefficients refrain from their oscillatory nature while presenting an exponential energy transfer. This is explained through the one-dimensional satisfaction of the Bragg resonance conditions by wave triads due to the non-dispersive propagation in this region even without depth changes. The energy evolution model is localized using a matching approach to account for both these behaviour types. The model is evaluated with respect to deterministic ensembles, field measurements and laboratory experiments while performing well in modelling monochromatic superharmonic self-interactions and infra-gravity wave generation from bichromatic waves and a realistic wave spectrum evolution. This lays physical and mathematical grounds for the validity of unexplained simplifications in former works and the capability to construct a formulation that consistently accounts for nonlinear energy transfers from deep to shallow water.

Large deviations for SPDEs of jump type

In this paper, we establish a large deviation principle for a fully nonlinear stochastic evolution equation driven by both Brownian motions and Poisson random measures on a given Hilbert space H. The weak convergence method plays an important role.

Stability of Neutral Impulsive Nonlinear Stochastic Evolution Equations with Time Varying Delays

AbstractIn this paper, we consider the stability of mild solutions to neutral impulsive nonlinear stochastic evolution equations with time varying delays. With the non‐Lipschitz condition, the Lipschitz condition being taken as a special case, on the impulsive term, the existence, uniqueness and sufficient conditions for the exponential stability in the pth moment and the almost sure exponential stability of the mild solutions are derived by employing a fixed point approach. An example is provided to illustrate the efficiency of the obtained theorems.

New travelling wave solutions for nonlinear stochastic evolution equations

The nonlinear stochastic evolution equations have a wide range of applications in physics, chemistry, biology, economics and finance from various points of view. In this paper, the \(\left({G^{\prime}}/{G}\right)\)-expansion method is implemented for obtaining new travelling wave solutions of the nonlinear (2 + 1)-dimensional stochastic Broer–Kaup equation and stochastic coupled Korteweg–de Vries (KdV) equation. The study highlights the significant features of the method employed and its capability of handling nonlinear stochastic problems.

Martingale Solution to Equations for Differential Type Fluids of Grade Two Driven by Random Force of Lévy Type

In this article we study a system of nonlinear non-parabolic stochastic evolution equations driven by Levy noise type. This system describes the motion of second grade fluids driven by random force. Global existence of a martingale solution is proved under general conditions on the noise. Since the coefficient of the noise does not satisfy a Lipschitz property, we could not prove any pathwise uniqueness result. We note that this is the first work dealing with a stochastic model for non-Newtonian fluids excited by external forces of Levy noise type.

Exponential stability of second-order stochastic evolution equations with Poisson jumps

This paper is concerned with the exponential stability problem of second-order nonlinear stochastic evolution equations with Poisson jumps. By using the stochastic analysis theory, a set of novel sufficient conditions are derived for the exponential stability of mild solutions to the second-order nonlinear stochastic differential equations with infinite delay driven by Poisson jumps. An example is provided to demonstrate the effectiveness of the proposed result.