Deterministic Parabolic Equations Research Articles

Unique continuation for a fourth-order stochastic parabolic equation

In this paper, we prove a boundary unique continuation property for a fourth-order stochastic parabolic equation evolving in a domain G⊂Rn. Our result shows that the value of the solution can be determined by the observation on an arbitrary open subset of the boundary. The quantitative version of this property can be derived by the global Carleman estimate, which is deduced from a weighted identity for a fourth-order stochastic parabolic operator. The results in this paper are also new even if the fourth-order stochastic parabolic equation reduces to the corresponding fourth-order deterministic parabolic equation.

Blowup solutions for stochastic parabolic equations

In this short paper, we are concerned with the blowup phenomenon of stochastic parabolic equations. By using the comparison principle and the results for deterministic parabolic equations, we obtain blowup results of solutions for stochastic parabolic equations.

On the convergence of stochastic transport equations to a deterministic parabolic one

A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension \(d\ge 2\); the case \(d=1\) is also investigated but no conclusive answer is obtained.

High-Order Operator Splitting for the Bidomain and Monodomain Models

The bidomain and monodomain models are multiscale reaction-diffusion models that describe the electrical activity in myocardial tissue. Because of the size and specific structure of the system required to produce meaningful data, numerical solutions are often found through the application of operator-splitting (OS) methods. First- and second-order OS methods have been successfully implemented for finding numerical solutions of the bidomain and monodomain models. It is well known that OS methods with order higher than two require backward time integration within each step. Accordingly, one may conclude that splitting methods with order higher than two are not suitable for models that contain deterministic parabolic equations because the necessary backward time integration would cause instabilities. In this paper, we demonstrate that it is indeed possible to obtain stable results from third-order OS methods applied to the bidomain and monodomain models. Furthermore, we demonstrate that the accompanying gain...

On some Gaussian Bernstein processes in and the periodic Ornstein–Uhlenbeck process

ABSTRACTIn this article, we prove new results regarding the existence of Bernstein processes associated with the Cauchy problem of certain forward–backward systems of decoupled linear deterministic parabolic equations defined in Euclidean space of arbitrary dimension , whose initial and final conditions are positive measures. We concentrate primarily on the case where the elliptic part of the parabolic operator is related to the Hamiltonian of an isotropic system of quantum harmonic oscillators. In this situation there are many Gaussian processes of interest whose existence follows from our analysis, including N-dimensional stationary and non-stationary Ornstein–Uhlenbeck processes, as well as Bernstein bridges which may be interpreted as Markovian loops in a particular case. We also introduce a new class of stationary non-Markovian processes, which we eventually relate to the N-dimensional periodic Ornstein–Uhlenbeck process, which is generated by a one-parameter family of non-Markovian probability measures. In this case, our construction requires the consideration of an infinite hierarchy of pairs of forward–backward heat equations associated with the pure point spectrum of the elliptic part, rather than just one pair in the Markovian case. We finally stress the potential relevance of these new processes to statistical mechanics, the random evolution of loops, and general pattern theory.

Global Carleman estimate for stochastic parabolic equations, and its application

This paper is addressed to proving a new Carleman estimate for stochastic parabolic equations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X. Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm. 5.2), one extra gradient term involving in that estimate is eliminated. Also, our improved Carleman estimate is established by virtue of the known Carleman estimate for deterministic parabolic equations. As its application, we prove the existence of insensitizing controls for backward stochastic parabolic equations. As usual, this insensitizing control problem can be reduced to a partial controllability problem for a suitable cascade system governed by a backward and a forward stochastic parabolic equation. In order to solve the latter controllability problem, we need to use our improved Carleman estimate to establish a suitable observability inequality for some linear cascade stochastic parabolic system, while the known Carleman estimate for forward stochastic parabolic equations seems not enough to derive the desired inequality.

Approximating Dynamics of a Singularly Perturbed Stochastic Wave Equation with a Random Dynamical Boundary Condition

This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting is used to establish the approximating equation of the system for a sufficiently small singular perturbation parameter. The approximating equation is a stochastic parabolic equation when the power exponent of the singular perturbation parameter is in $[1/2, 1)$ but is a deterministic wave equation when the power exponent is in $(1, +\infty)$. Moreover, if the power exponent of a singular perturbation parameter is bigger than or equal to $1/2$, the same limiting equation of the system is derived in the sense of distribution, as the perturbation parameter tends to zero. This limiting equation is a deterministic parabolic equation.

Higher-order operator splitting methods for deterministic parabolic equations

The Sheng-Suzuki theorem states that all exponential operator splitting methods of order greater than 2 must contain negative time integration. There have been claims in the literature that higher-order splitting methods for deterministic parabolic equations are unstable due to this fact. We show stability for a class of higher-order splitting methods for integrating deterministic parabolic equations. We note that problems with backwards time integration will still exist for stochastic integration methods for which information is lost and backward timesteps become ill-defined. Therefore, completely positive splitting methods, such as those developed by Chin, still have an important place. We present numerical results from first-, second-, third- and fourth-order methods showing that the error becomes increasingly small as the order increases.

Fractal dimension of a random invariant set

In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.

Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.

On the Stochastic Burgers’ Equation in the Real Line

In this paper we establish the existence and uniqueness of an $L^2(\mathbb{R})$ -valued solution for a one-dimensional Burgers’ equation perturbed by a space–time white noise on the real line. We show that the solution is continuous in space and time, provided the initial condition is continuous. The main ingredients of the proof are maximal inequalities for the stochastic convolution, and some a priori estimates for a class of deterministic parabolic equations.