On the trace embedding and its applications to evolution equations

Agresti, Antonio; Lindemulder, Nick; Veraar, Mark

doi:10.48550/arxiv.2104.05063

Cited by 2 publications

(12 citation statements)

References 42 publications

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“…As in [6,Theorem 2.7], by further bootstrapping, one can even get θ 2 ∈ (0, 2) in (2). Moreover, this can be further extended by (for instance) applying Schauder theory.…”

Section: Illustrationmentioning

confidence: 94%

“…For instance, in case X Tr κ, p is not critical, for this one needs a suitable a priori bound for the solution which implies the existence of the limit lim t↑σ u(t) in the "trace space" X Tr κ, p on the set {σ < ∞}. According to Theorem 1.2 (2) this can only happen if P(σ < ∞) = 0, and thus σ = ∞ a.s. Similar considerations hold for Theorem 1.2 (1) and (3).…”

Section: Blow-up Criteriamentioning

confidence: 99%

“…Some further explanations what this means can be found below. Using (2), the proof of (3) follows by deriving an energy estimate from Itô's formula and applying the blow-up criterium of Theorem 1.3 (2).…”

Section: Illustrationmentioning

confidence: 99%

“…where in case (2) To conclude, let us recall the trace embedding. Here in the limiting case we write H 1, p = W 1, p and 0 H 1, p = 0 W 1, p .…”

Section: Weighted Function Spacesmentioning

confidence: 99%

“…The above result follows from [4,35] (see also [3,Proposition 2.10]) provided X 1 = D(A) and A is a sectorial operator on X . The more general case will be considered in [2], but is not needed here.…”

Section: Weighted Function Spacesmentioning

confidence: 99%

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Nonlinear parabolic stochastic evolution equations in critical spaces part II

Agresti

Veraar

2022

J. Evol. Equ.

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This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical $$L^2$$ L 2 -settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.

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“…As in [6,Theorem 2.7], by further bootstrapping, one can even get θ 2 ∈ (0, 2) in (2). Moreover, this can be further extended by (for instance) applying Schauder theory.…”

Section: Illustrationmentioning

confidence: 94%

Section: Blow-up Criteriamentioning

confidence: 99%

Section: Illustrationmentioning

confidence: 99%

“…where in case (2) To conclude, let us recall the trace embedding. Here in the limiting case we write H 1, p = W 1, p and 0 H 1, p = 0 W 1, p .…”

Section: Weighted Function Spacesmentioning

confidence: 99%

Section: Weighted Function Spacesmentioning

confidence: 99%

See 3 more Smart Citations

Nonlinear parabolic stochastic evolution equations in critical spaces part II

Agresti

Veraar

2022

J. Evol. Equ.

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Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence*

Agresti

Veraar

2022

Nonlinearity

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In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an L p -setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers’ equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.

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On the trace embedding and its applications to evolution equations

Cited by 2 publications

References 42 publications

Nonlinear parabolic stochastic evolution equations in critical spaces part II

Nonlinear parabolic stochastic evolution equations in critical spaces part II

Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence*

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