Nonlinear Stochastic Partial Differential Equations Research Articles

On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.

Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by the total variation flow, respectively, by a singular p-Laplace-type operator. We impose nonlinear zero Neumann boundary conditions and precisely investigate their connection with the coefficient fields of the noise. This solves an open problem posed in Barbu et al. (2013) [7] and Barbu and Röckner (2015) [10].

Shear melting at the crystal-liquid interface: Erosion and the asymmetric suppression of interface fluctuations.

The influence of an applied shear on the planar crystal-melt interface is modeled by a nonlinear stochastic partial differential equation of the interface fluctuations. A feature of this theory is the asymmetric destruction of interface fluctuations due to advection of the crystal protrusions on the liquid side of the interface only. We show that this model is able to qualitatively reproduce the nonequilibrium coexistence line found in simulations. The impact of shear on spherical clusters is also addressed.

Nonlinear parabolic SPDEs involving Dirichlet operators

We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet forms. In proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.

Solution of the stochastic generalized shallow-water wave equation using RVT technique

In this paper, some exact solutions of the stochastic generalized nonlinear shallow-water wave (SGNSWW) equation are obtained. This equation is an important equation in fluid mechanics field. Opposite to what is usually assumed in the literature, the coefficients of the nonlinear terms in this stochastic nonlinear partial differential equation (SNLPDE) are considered to be random quantities. The random variable transformation (RVT) technique is combined with the modified extended-tanh function method (METFM) to get the stochastic solutions represented by the probability density functions (PDFs) of the solution processes in terms of the PDFs of the random coefficients. These solutions are illustrated graphically along the spacial and time dimensions at a certain wave speed.

NUMERICAL SIMULATIONS OF A NONLINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATION MODELING PHYTOPLANKTON AGGREGATION

In this paper, we are interested in the numerical simulation of a nonlinear stochastic partial differential equation (SPDE) arising as a model of phytoplankton aggregation. This SPDE consists of a diffusion equation with a chemotaxis term and a branching noise. We develop and implement a numerical scheme to solve this SPDE and present its numerical solutions for parameter values corresponding to real conditions in nature. Further, a comparison is made with two deterministic versions of the SPDE, that are advection–diffusion equations with linear and nonlinear reaction terms, to emphasize the efficiency of the stochastic equation in modeling the aggregation behavior in phytoplankton.

Numerical methods for high-dimensional probability density function equations

In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.

A Wong-Zakai theorem for stochastic PDEs

We prove a version of the Wong-Zakai theorem for one-dimensional parabolic nonlinear stochastic PDEs driven by space-time white noise. As a corollary, we obtain a detailed local description of solutions. Dedicated to the memory of Kiyosi It\^o on the occasion of the 100th anniversary of his birth.

New analytical approach for transition to slow 3-D turbulence

Analytical non-perturbative study of the three-dimensional nonlinear stochastic partial differential equation with additive thermal noise analogous to that proposed by Nikolaevskii V.N. to describe longitudinal seismic waves is presented. The equation has a threshold of short-wave instability and symmetry providing long wave dynamics. New mechanism of quantum chaos generating in nonlinear dynamical systems with infinite number of degrees of freedom is proposed. The hypothesis says that physical turbulence could be identified with quantum chaos of considered type. It is shown that the additive thermal noise destabilizes dramatically the ground state of the Nikolaevskii system causing it to make a direct transition from a spatially uniform to a turbulent state.

Non-Linear Time-Advanced Backward Stochastic Partial Differential Equations With Jumps

We prove an existence and uniqueness result for non-linear time-advanced backward stochastic partial differential equations with jumps (ABSPDEJs). We then apply our results to study a time-advanced backward type of stochastic generalized porous medium equations with jumps.

Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations

In this paper we initiate the mathematical analysis of a system of nonlinear Stochastic Partial Differential equations describing the motion of turbulent Non-Newtonian media in the presence of fluctuating magnetic field. The system is basically obtained by a coupling of the dynamical equations of a Non-Newtonian fluids having $p$-structure and the Maxwell equations. We mainly show the existence of weak martingale solutions and their exponential decay when time goes to infinity.

Approximate Solution of Stochastic Partial Differential Equation with Random Neumann Boundary Condition

In this paper we approximate the solution of a parabolic nonlinear stochastic partial differential equation (SPDE) with cubic nonlinearity and with random Neumann boundary condition via a stochastic ordinary differential equation (SODE) which is a stochastic amplitude equation near a change of stability.

New Scenario for Transition to Slow 3-D Turbulence

Analytical non-perturbative study of the three-dimensional nonlinear stochastic partial differential equation with additive thermal noise, analogous to that proposed by V.N. Nikolaevskii [1]-[5]to describe longitudinal seismic waves, is presented. The equation has a threshold of short-wave instability and symmetry, providing long wave dynamics. New mechanism of quantum chaos generating in nonlinear dynamical systems with infinite number of degrees of freedom is proposed. The hypothesis is said, that physical turbulence could be identified with quantum chaos of considered type. It is shown that the additive thermal noise destabilizes dramatically the ground state of the Nikolaevskii system thus causing it to make a direct transition from a spatially uniform to a turbulent state.

Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions

We study the stochastic Allen-Cahn equation driven by a noise term with intensity $\sqrt{\varepsilon}$ and correlation length $\delta$ in two and three spatial dimensions. We study diagonal limits $\delta, \varepsilon \to 0$ and describe fully the large deviation behaviour depending on the relationship between $\delta$ and $\varepsilon$. The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length $\delta$ and fixed noise intensity $\varepsilon$. One key fact is that in order to get non-trivial limits as $\delta \to 0$, it is necessary to introduce diverging counterterms. The theory of regularity structures allows to rigorously analyse this renormalisation procedure for a number of interesting equations. Our main result is a large deviation principle for these renormalised solutions. One interesting feature of this result is that the diverging renormalisation constants disappear at the level of the large deviations rate function. We apply this result to derive a sharp condition on $\delta, \varepsilon$ that guarantees a large deviation principle for diagonal schemes $\varepsilon, \delta \to 0$ for the equation without renormalisation.

Exponential convergence of non-linear monotone SPDEs

For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\lambda>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that $$ \sup_{\mu(f^2)\le 1}||P_tf-\mu(f)||_\infty \le C e^{-\lambda t}, t\ge 1.$$ By using the coupling by change of measure in the line of [F.-Y. Wang, Ann. Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential convergence rate are derived for a class of non-linear monotone stochastic partial differential equations. The main result is illustrated by the stochastic porous medium equation and the stochastic $p$-Laplace equation respectively. Finally, the $V$-uniformly exponential convergence is investigated for stochastic fast-diffusion equations.

Ensemble Averaging for Dynamical Systems Under Fast Oscillating Random Boundary Conditions

This article is devoted to providing a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable “mixing” conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial differential equation, with the deviation process (i.e., the approximation error process) quantified as the solution of a linear stochastic partial differential equation.

Practical exponential stability in mean square of stochastic partial differential equations

The main aim of this paper is to establish some criteria for the mean square and almost sure practical exponential stability of a nonlinear monotone stochastic partial differential equations.

Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

In this article, we will prove the existence, uniqueness and Holder regularity of the solution to the fractional stochastic partial differential equation of the form $$\begin{aligned} \frac{\partial }{\partial t}u(t,x)=\mathfrak {D}(x,D)u(t,x)+\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^2 W^H}{\partial t\partial x}(t,x), \end{aligned}$$ where \(\mathfrak {D}(x,D)\) denotes the Markovian generator of stable-like Feller process, \(f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) is a measurable function, and \(\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)\) is a double-parameter fractional noise. In addition, we establish lower and upper Gaussian bounds for the probability density of the mild solution via Malliavin calculus and the new tool developed by Nourdin and Viens (Electron J Probab 14:2287–2309, 2009).

Preface Issue 3-2014

Is it possible to model human societies and to predict on this basis some characteristic features of their long term development? This question is addressed by F. Thomas Bruss’s survey article entitled “Grenzen einer jeden Gesellschaft”. The starting point are two widely acceptable hypotheses governing the decisions of human beings. The modeling is done by means of branching processes, which depend on the available resources and the chosen politics. The main result is an enclosure theorem which states that there are two extreme forms of societies (weakest-first and strongest-first) which build an envelope for all possible societies within the underlying model. The obituary for Wolfgang Schwarz, written by Karl-Heinz Indlekofer, Lutz G. Lucht and Jorn Steuding, reviews some of his most important results. His œuvre is focussed on the boundary behaviour of power series, Tauberian theorems, sieve methods and, mainly, on arithmetic functions. His monograph “Arithmetical functions” with Jurgen Spilker is a standard reference in this field of number theory. The Deutsche Mathematiker-Vereinigung is grateful to Wolfgang Schwarz for his commitment, among other things he served as its chairman in 1986 and 1987. In collaboration with the Zentralblatt fur Mathematik, we have in each issue a contribution to the category “Classics Revisited”. Here, Niels Jacob reviews again Lars Hormander’s encyclopedic series of four monographs on “The analysis of linear partial differential operators”. This review describes in some detail the contents and the huge influence of these books of the 1962 Fields Medalist. It is, however, not intended to serve as a kind of obituary on Lars Hormander, who died on November 25, 2012. Something of this type is for example Nicolas Lerner’s “Tribute to Lars Hormander”, which appeared in the June 2013-EMS-Newsletter (No. 88) and is available online also at http://www.math.jussieu.fr/ lerner/. The recent books, which are reviewed in the current issue, are concerned with “The Mathematics of Planet Earth”, Markov numbers – a topic from diophantine equations, nonlinear stochastic partial differential equations, and the topology of algebraic curves.

Jie Xiong: “Three Classes of Nonlinear Stochastic Partial Differential Equations”

Jie Xiong: “Three Classes of Nonlinear Stochastic Partial Differential Equations”