Impacts of noise on a class of partial differential equations

Lv, Guangying; Duan, Jinqiao

doi:10.1016/j.jde.2014.12.002

Cited by 29 publications

(35 citation statements)

References 25 publications

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“…In particular, there have been many works devoted to the analysis of the ultimate behavior of solutions to stochastic partial differential equations of parabolic type which specifically occur in population dynamics, population genetics, nerve pulse propagation and related topics, to name only a few (see, e.g., [19] for a brief account of some of those works and the references therein). Moreover, there have also been several more recent articles dealing with the analysis of solutions to various types of semilinear parabolic stochastic partial differential equations driven either by a Brownian noise, or by a fractional noise with Hurst parameter H ∈ 1 2 , 1 (see, e.g., [3], [4], [5]- [7], and the plethora of references therein, particularly [11]). While these works have been primarily centered around questions of global existence, uniqueness and blowup in finite time, there have also been investigations essentially motivated by issues in financial mathematics devoted to the analysis of problems that involve a mixture of a Brownian noise with a fractional noise, within the realm of both ordinary and partial stochastic differential equations (see, e.g., [9], [13]- [15] and the references therein).…”

Section: Outline and Statement Of The Main Resultsmentioning

confidence: 99%

Global variational solutions to a class of fractional SPDE’s on unbounded domains

Dozzi

Touibi

Vuillermot

2018

Stochastic Analysis and Applications

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In this article we prove new results regarding the existence and the uniqueness of global variational solutions to Neumann initial-boundary value problems for a class of non-autonomous stochastic parabolic partial differential equations. The equations we consider are defined on unbounded open domains in Euclidean space satisfying certain geometric conditions, and are driven by a multiplicative noise derived from an infinite-dimensional fractional Wiener process characterized by a sequence of Hurst parameters H=(Hi) i∈N + ⊂ 1 2 , 1 . These parameters are in fact subject to further constraints that are intimately tied up with the nature of the nonlinearity in the stochastic term of the equations, and with the choice of the functional spaces in which the problem at hand is well-posed. Our method of proof rests on compactness arguments in an essential way.

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Section: Outline and Statement Of The Main Resultsmentioning

confidence: 99%

Global variational solutions to a class of fractional SPDE’s on unbounded domains

Dozzi

Touibi

Vuillermot

2018

Stochastic Analysis and Applications

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“…For example, if the determinisitc PDE part is a cross-diffusion system with an entropy structure [37] and if the noise is multiplicative, we expect that the maximal local pathwise mild solution obtained in Theorem 3.28 is indeed a global one. We plan to investigate this in a future work using for instance using Khashminski's test for non-explosion; see for example [64,Lemma 4.1], [16,Theorem 3.2] or [46,Section 5].…”

Section: Remark 329mentioning

confidence: 99%

Pathwise mild solutions for quasilinear stochastic partial differential equations

Kuehn

Neamţu

2020

Journal of Differential Equations

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Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, includingamong others -cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time.Here we aim to develop a theory for quasilinear stochastic evolution equations (1.1) using a semigroup approach. The main theme is to extend the very general deterministic theory of quasilinear Cauchy problems [3,61,63]. The key idea is to employ a modified definition of mild solutions [57] for the quasilinear case in comparison to the more classical parabolic SPDE setting [54]. Before we describe our approach in more detail, we briefly review some other techniques and solution concepts used for (certain subclasses of) the SPDE (1.1). Instead of mild solutions, one may instead use weak, or martingale, solutions [13,20,24,35] of (1.1); here weak solution is interpreted in the classical PDE sense while these solutions are also sometimes referred to as strong solutions from a probabilistic perspective [55]. There are also several works exploiting the additional assumption of monotone coefficients [43] particularly in the case of the stochastic porous medium equation [9,10], where A(u) = ∆(a(u)) for a maximal monotone map a and F ≡ 0. Other approaches to quasilinear SPDEs are based upon a gradient structure [29], approximation methods [38,39], kinetic solutions [20,30], or directly looking at strong (in the PDE sense) solutions [36].One may ask, why one might want to prove the existence of pathwise mild solutions obtained by a suitable variations-of-constants/Duhamel formula [34] instead of working with weak solutions obtained in a formulation via test functions [26]? One reason is that a mild formulation is often more natural to work with in the context of (random) dynamical systems for SPDEs [18,34]. In fact, many classical results regarding dynamics and long-time behavior of semilinear SPDEs are often crucially based upon the mild formulation and semigroups [34]. We expect this theory to generalize a lot easier also in the quasilinear case if one does not have to work with weak(er) solutions. If we take the SKT system again as a motivation, then there are deterministic results regarding the existence of attractors using weak [53] as well as mild [62] solutions concepts. We intend to investigate the existence of random attractors for the stochastic SKT equation ...

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“…We remark that the nonlinear term g satisfying the Lipschitz condition is natural. It follows from [26] that if g satisfies the local Lipschitz condition, then the solution of problem (2.1) maybe blow up in finite time.…”

Section: Entropy Solutionmentioning

confidence: 99%

Heterogeneous stochastic scalar conservation laws with non-homogeneous Dirichlet boundary conditions

2018

J. Hyper. Differential Equations

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We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function.

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Impacts of noise on a class of partial differential equations

Cited by 29 publications

References 25 publications

Global variational solutions to a class of fractional SPDE’s on unbounded domains

Global variational solutions to a class of fractional SPDE’s on unbounded domains

Pathwise mild solutions for quasilinear stochastic partial differential equations

Heterogeneous stochastic scalar conservation laws with non-homogeneous Dirichlet boundary conditions

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