Stochastically Forced Compressible Fluid Flows

Breit, Dominic; Feireisl, Eduard; Hofmanová, Martina

doi:10.1515/9783110492552

Cited by 80 publications

(102 citation statements)

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“…For the proof of Theorem 2.1 one can follow the ideas in e.g. [2][3][4]16,19,29,48] by constructing a sequence of approximations for a problem with cut-off for the W 1,∞ -norm. Such a cut-off implies at-most linear growth of u and guarantees the global existence of an approximate solution.…”

Section: Main Results and Remarksmentioning

confidence: 99%

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On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

Rohde

Tang

2020

Nonlinear Differ. Equ. Appl.

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We consider a class of stochastic evolution equations that include in particular the stochastic Camassa–Holm equation. For the initial value problem on a torus, we first establish the local existence and uniqueness of pathwise solutions in the Sobolev spaces $$H^s$$ H s with $$s>3/2$$ s > 3 / 2 . Then we show that strong enough nonlinear noise can prevent blow-up almost surely. To analyze the effects of weaker noise, we consider a linearly multiplicative noise with non-autonomous pre-factor. Then, we formulate precise conditions on the initial data that lead to global existence of strong solutions or to blow-up. The blow-up occurs as wave breaking. For blow-up with positive probability, we derive lower bounds for these probabilities. Finally, the blow-up rate of these solutions is precisely analyzed.

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Section: Main Results and Remarksmentioning

confidence: 99%

“…This will be essential to pass to the limit when establishing the existence of a martingale solution as an intermediate step, cf. [3,48,49]. • The uniform-in-time assumption (2.2) bounds the growth of the L 2 (U ; H s )-norm of the noise coefficient in terms of a product of a nonlinear function of the W 1,∞ -norm and the H s -norm.…”

Section: Assumptionsmentioning

confidence: 99%

On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

Rohde

Tang

2020

Nonlinear Differ. Equ. Appl.

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“…Remark 2.1 We remark here that F 3 (u) in (1.6) will disappear when k = 1. The proof for Theorem 2.1 combines the techniques as employed in the papers [3][4][5]18,20,29,58]. However, the Faedo-Galerkin method used e.g.…”

Section: Resultsmentioning

confidence: 99%

On a Stochastic Camassa–Holm Type Equation with Higher Order Nonlinearities

Rohde

Tang

2020

J Dyn Diff Equat

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The subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces H s with s > 3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.

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“…In order to find an appropriate notion of renormalized solutions to (1), we prove an Itô formula in the L 1 -framework. We remark that the combined Itô chain and product rule from [10], Appendix A4 does not apply to our situation for two reasons. Firstly, we take the bouded domain D ⊂ R d into account in our regularizing procedure by adding a cutoff function (see Appendix, Subsection 9.1).…”

Section: Contraction Principlementioning

confidence: 99%

“…Evolution equations of p-Laplace type may appear as continuity equations in the study of gases flowing in pipes of uniform cross sectional areas and in models of filtration of an incompressible fluid through a porous medium (see [3,14]): In the case of a turbulent regime, a nonlinear version of the Darcy law of p-power law type for 1 < p < 2 is more appropriate (see [14]). Turbulence is often associated with the presence of randomness (see [10] and the references therein). Adding random influences to the model, we also take uncertainties and multiscale interactions into account.…”

Section: 1mentioning

confidence: 99%

Well-posedness of renormalized solutions for a stochastic p -Laplace equation with L^1 -initial data

Sapountzoglou¹,

Zimmermann²

2021

Discrete &Amp; Continuous Dynamical Systems - A

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We consider a p-Laplace evolution problem with multiplicative noise on a bounded domain D ⊂ R d with homogeneous Dirichlet boundary conditions for 1 < p < ∞. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic p-Laplace equations with L 1-initial data and study existence and uniqueness of solutions in this framework.

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Stochastically Forced Compressible Fluid Flows

Cited by 80 publications

References 0 publications

On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

On the stochastic Dullin–Gottwald–Holm equation: global existence and wave-breaking phenomena

On a Stochastic Camassa–Holm Type Equation with Higher Order Nonlinearities

Well-posedness of renormalized solutions for a stochastic p -Laplace equation with L^1 -initial data

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