Nonlinear Stochastic Partial Differential Equations Research Articles

Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle

In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis and PDEs). Secondly, we show the pathwise uniqueness of the solution in a 2D domain. In contrast to several works in the deterministic setting we replace the Ginzburg–Landau function mathbb {1}_{|{mathbf {n}}|le 1}(|{mathbf {n}}|^2-1){mathbf {n}} by an appropriate polynomial f({mathbf {n}}) and we give sufficient conditions on the polynomial f for these two results to hold. Our third result is a maximum principle type theorem. More precisely, if we consider f({mathbf {n}})=mathbb {1}_{|d|le 1}(|{mathbf {n}}|^2-1){mathbf {n}} and if the initial condition {mathbf {n}}_0 satisfies |{mathbf {n}}_0|le 1, then the solution {mathbf {n}} also remains in the unit ball.

Time integration of stochastic generalized equations of motion using SSFEM

The paper develops an integration approach to stochastic nonlinear partial differential equations (SPDE’s) with parameters to be random fields. The methodology is based upon assumption that random fields are from a special class of functions, and can be described as a product of two functions with dependent and independent random variables. Such an approach allows one to use Karhunen-Lo`eve expansion directly, and the modified stochastic spectral finite element method (SSFEM). It is assumed that a random field is stationary and Gaussian while the autocovariance function is known. A numerical example of onedimensional heat waves analysis is shown.

Interface Liquid-Solid: Global Bifurcation Multiparameter of Flow Water Waves through Porous Compacted Granular by andldquo;Yakam Matrixandrdquo;

The construction of the tourbillon balls is fundamentally well known to Constatin and Strauss. In this work, we apply the flow of water through the compacted granular, in order to arrive at a model of passage in confinement and stagnation through the pores. An approach agrees of the advanced theory of water waves and the application of dynamics in microfluidics is discussed to improve “ASEAD Project water for everybody”. The approximation solution of nonlinear stochastic partial differential equation for only element of “Yakam Matrix” liquid-solid interface by simple understanding bifurcation behavior will follow the some trajectory of waves of water, granular compaction and dynamics of fluids at all scales, then we are still just at the beginning.

A Regularized Dean--Kawasaki Model: Derivation and Analysis

The Dean--Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, driven by space-t...

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

Irreducibility and strong Feller property for non-linear SPDEs

ABSTRACTUnder some non-degeneracy condition, the strong Feller property and irreducibility are studied for non-linear stochastic partial differential equations driven by multiplicative noise within the framework called ‘variational approach’. Our result for irreducibility can be applied to equations with locally monotone coefficients. In some special cases, we discuss the Hölder continuity of the associated Markov semigroups. The main results are applied to several examples such as stochastic Burgers equation, stochastic porous media equation and stochastic fast diffusion equation.

Centre manifolds for infinite dimensional random dynamical systems

ABSTRACTStochastic centre manifolds theory are crucial in modelling the dynamical behaviour of complex systems under stochastic influences. The existence of stochastic centre manifolds for infinite dimensional random dynamical systems is shown under the assumption of exponential trichotomy. The theory provides a support for the discretisations of nonlinear stochastic partial differential equations with space–time white noise.

On the Solution of Stochastic Generalized Burgers Equation

We are interested in one dimensional nonlinear stochastic partial differential equation: the generalized Burgers equation with homogeneous Dirichlet boundary conditions, perturbed by additive space-time white noise. We propose a result of existence and uniqueness of the local solution to the viscous equation using fixed point argument, then if we impose a condition to the viscosity coefficient we can prove that this solution is global.

Probabilistic interpretation for solutions of fully nonlinear stochastic PDEs

In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between the 2BDSDEs and a class of parabolic fully nonlinear Stochastic PDEs. Precisely, we show that the Markovian solution of 2BDSDEs provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear SPDEs.

Error estimates of finite element methods for nonlinear fractional stochastic differential equations

In this paper, we consider the Galerkin finite element approximations of the initial value problem for the nonlinear fractional stochastic partial differential equations with multiplicative noise. We study a spatial semidiscrete scheme with the standard Galerkin finite element method and a fully discrete scheme based on the Goreno–Mainardi–Moretti–Paradisi (GMMP) scheme. We establish strong convergence error estimates for both semidiscrete and fully discrete schemes.

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate.

Particle representations for stochastic partial differential equations with boundary conditions

In this article, we study a weighted particle representation for a class of stochastic partial differential equations with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations (SDEs). The evolution of the particles is modeled by an infinite system of stochastic differential equations with reflecting boundary condition and driven by independent finite dimensional Brownian motions. The weights of the particles evolve according to an infinite system of stochastic differential equations driven by a common cylindrical noise $W$ and interact through $V$, the associated weighted empirical measure. When the particles hit the boundary their corresponding weights are assigned a pre-specified value. We show the existence and uniqueness of a solution of the infinite dimensional system of stochastic differential equations modeling the location and the weights of the particles. We also prove that the associated weighted empirical measure $V$ is the unique solution of a nonlinear stochastic partial differential equation driven by $W$ with Dirichlet boundary condition. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong.

Stochastic evolution equations with Wick-polynomial nonlinearities

We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.

Approximate the dynamical behavior for stochastic systems by Wong–Zakai approaching

Random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. In a general way, these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds and foliations. After briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises, the corresponding Wong–Zakai type of convergence result in path-wise sense is established.

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.

Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form dXt−divγ(∇Xt)dt+β(Xt)dt∋B(t,Xt)dWt, where γ and β are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on Rd and R respectively, and W is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray–Lions conditions on γ and with no restrictive smoothness or growth assumptions on β. The operator B is assumed to be Hilbert–Schmidt and to satisfy some classical Lipschitz conditions in the second variable.

Estimates of blow‐up times of a system of semilinear SPDEs

In this paper, blow‐up property to a system of nonlinear stochastic PDEs driven by two‐dimensional Brownian motions is investigated. The lower and upper bounds for blow‐up times are obtained. When the system parameters satisfy certain conditions, the explicit solutions of a related system of random PDEs are deduced, which allows us to use Yor's formula to obtain the distribution functions of several blow‐up times. Particularly, the impact of noises on the life span of solutions is studied as the system parameters satisfy different conditions. Copyright © 2016 John Wiley & Sons, Ltd.

Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations

This paper is concerned with a class of stochastic Hamilton–Jacobi–Bellman equations with controlled leading coefficients, which are fully nonlinear backward stochastic partial differential equations (BSPDEs for short). In order to formulate the weak solution for such kind of BSPDEs, a class of regular random parabolic potentials are introduced in the backward stochastic framework. The existence and uniqueness of weak solution is proved, and for the partially non-Markovian case, we obtain the associated gradient estimate. As a byproduct, the existence and uniqueness of solution for a class of degenerate reflected BSPDEs is discussed as well.

Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises

We study a class of abstract nonlinear stochastic equations of hyperbolic type driven by jump noises, which covers both beam equations with nonlocal, nonlinear terms and nonlinear wave equations. We derive an Ito formula for the local mild solution which plays an important role in the proof of our main results. Under appropriate conditions, we prove the non-explosion and the asymptotic stability of the mild solution.

Periodic solutions of Wick-type stochastic Korteweg–de Vries equations

Nonlinear stochastic partial differential equations have a wide range of applications in science and engineering. Finding exact solutions of the Wick-type stochastic equation will be helpful in the theories and numerical studies of such equations. In this paper, Kudrayshov method together with Hermite transform is implemented to obtain exact solutions of Wick-type stochastic Korteweg–de Vries equation. Further, graphical illustrations in two- and three-dimensional plots of the obtained solutions depending on time and space are also given with white noise functionals.