Finite Speed of Propagation for Stochastic Porous Media Equations

Gess, Benjamin

doi:10.1137/120894713

Cited by 16 publications

(12 citation statements)

References 37 publications

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“…the controllability has been proved for strictly parabolic equations (even partial differential equations). Here, we shall prove controllability for a porous media type equation (which is an example of a degenerate parabolic equation; see [9,24]):…”

Section: Notions Notations and Formulation Of The Resultsmentioning

confidence: 99%

Global Controllability for Quasilinear Non-negative Definite System of ODEs and SDEs

Djordjevic¹,

Konjik²,

Mitrović³

et al. 2021

Preprint

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We consider exact and averaged control problem for a system of quasi-linear ODEs and SDEs with a non-negative definite symmetric matrix of the system. The strategy of the proof is the standard linearization of the system by fixing the function appearing in the nonlinear part of the system, and then applying the Leray-Schauder fixed point theorem.We shall also need the continuous induction arguments to prolong the control to the final state which is a novel approach in the field. This enables us to obtain controllability for arbitrarily large initial data (so called global controllability).

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Section: Notions Notations and Formulation Of The Resultsmentioning

confidence: 99%

Global Controllability for Quasilinear Non-negative Definite System of ODEs and SDEs

Djordjevic¹,

Konjik²,

Mitrović³

et al. 2021

Preprint

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“…where ξ denotes space-time white noise. The nonlinear character of ( 1) is that of a fully nonlinear equation rather than a quasi-linear equation, since rewriting (1) as the quasi-linear equation (7) is not helpful as we explain below, and since the deterministic estimates we need are related to the linearization of a fully nonlinear equation, cf. (15), rather than to the linearization of a quasi-linear equation (this distinction would be more pronounced in a multidimensional case).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Sometimes, structural assumptions allow to mimic an approach that is obvious in the semi-linear case, namely the approach of decomposing the solution into a rough part w that solves a more explicitly treatable stochastic differential equation and a more regular part v that solves a parabolic equation with random coefficients and/or right-hand-side described through w, and then allows for an application of deterministic regularity theory. We refer to [7] for an example with a multiplicative decomposition of this type. The recent work by Debussche et.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Hölder regularity for a non-linear parabolic equation driven by space-time white noise

Otto,

Weber

2015

Preprint

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We consider the non-linear equation T −1 u + ∂ t u − ∂ 2 x π(u) = ξ driven by space-time white noise ξ, which is uniformly parabolic because we assume that π ′ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of π ′ we show that the stationary solution is -as for the linear case -almost surely Hölder continuous with exponent α for any α < 1 2 w. r. t. the parabolic metric. More precisely, we show that the corresponding local Hölder norm has stretched exponential moments. On the stochastic side, we use a combination of martingale arguments to get second moment estimates with concentration of measure arguments to upgrade to Gaussian moments. On the deterministic side, we first perform a Campanato iteration based on the De Giorgi-Nash Theorem as well as finite and infinitesimal versions of the H −1 -contraction principle, which yields Gaussian moments for a weaker Hölder norm. In a second step this estimate is improved to the optimal Hölder exponent at the expense of weakening the integrability to stretched exponential.

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“…In the stochastic setting, the techniques of [19] have been generalized by [14,25] to provide sufficient criteria for the occurrence of waiting time phenomena and for qualitative results on finite speed of propagation for stochastic p-Laplace and stochastic porous-media equations. For finite speed of propagation for the latter equations, we also mention [2,16] which use different techniques.…”

Section: Introductionmentioning

confidence: 99%

Zero-contact angle solutions to stochastic thin-film equations

Grün¹,

Lorenz²

2021

Preprint

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We establish existence of nonnegative martingale solutions to stochastic thin-film equations with compactly supported initial data under Stratonovich noise. Based on so called α-entropy estimates, we show that almost surely these solutions are classically differentiable in space almost everywhere in time and that their derivative attains the value zero at the boundary of the solution's support. I.e., from a physics perspective, they exhibit a zero-contact angle at the three-phase contact line between liquid, solid, and ambient fluid. These α-entropy estimates are first derived for almost surely strictly positive solutions to a family of stochastic thin-film equations augmented by second-order linear diffusion terms. Using Itô's formula together with stopping time arguments, the Jakubowski/Skorokhod calculus, and martingale identification techniques, the passage to the limit of vanishing regularization terms gives the desired existence result.

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Finite Speed of Propagation for Stochastic Porous Media Equations

Cited by 16 publications

References 37 publications

Global Controllability for Quasilinear Non-negative Definite System of ODEs and SDEs

Global Controllability for Quasilinear Non-negative Definite System of ODEs and SDEs

Hölder regularity for a non-linear parabolic equation driven by space-time white noise

Zero-contact angle solutions to stochastic thin-film equations

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