The Navier-Stokes Equations

Sohr, Hermann

doi:10.1007/978-3-0348-8255-2

Cited by 273 publications

(129 citation statements)

References 48 publications

(99 reference statements)

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“…where The operator A is boundedly invertible (see [15] [81] for the details) that, by applying the Helmholtz projection P to u, the Navier-Stokes equation (8.1) can be reformulated as an abstract stochastic evolution on…”

Section: Discussionmentioning

confidence: 99%

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Neerven¹,

Veraar²,

Weis³

2012

SIAM J. Math. Anal.

115

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Abstract. We prove maximal L p -regularity for the stochastic evolution equation dU (t) + AU (t) dt = F (t, U (t)) dt + B(t, U (t)) dW H (t), t ∈ [0, T ], U (0) = u 0 , under the assumption that A is a sectorial operator with a bounded H ∞ -calculus of angle less than 1. Introduction. Maximal L p -regularity techniques have been pivotal in much of the recent progress in the theory of parabolic evolution equations (see [2,22,25,54,76,86] and the references therein). Among other things, such techniques provide a systematic and powerful tool to study nonlinear and time-dependent parabolic problems.For stochastic parabolic evolution equations, maximal L p -regularity results have been obtained previously by Krylov for second order problems on R d [44,46,47,48,49], by Kim for second order problems on bounded domains in R d [43], and by Mikulevicius and Rozovskii for Navier-Stokes equations [63]. A systematic theory of maximal L p -regularity for stochastic evolution equations, however, based on abstract operator-theoretic properties of the operators governing the equation, has yet to be developed. A first step towards such a theory has been taken in our recent paper [68], where it was shown that if A is a sectorial operator with a bounded H ∞ -calculus of angle <

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Section: Discussionmentioning

confidence: 99%

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Neerven¹,

Veraar²,

Weis³

2012

SIAM J. Math. Anal.

115

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“…Assume for the moment that there exists R > 0 such that v C is supported on B.0; R/. Proceeding as in Sohr [45], for example, an energy estimate gives…”

Section: Linear Stabilitymentioning

confidence: 99%

On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions

Germain

Ghoul

Miura

2017

Comm Pure Appl Math

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“…The corresponding Helmholtz projection, ie, the L 2 -orthogonal projection onto L 2 (Ω), is denoted by P , cf, eg, Sohr. 26 Let…”

Section: Preliminariesmentioning

confidence: 99%

Weak solutions for a diffuse interface model for two‐phase flows of incompressible fluids with different densities and nonlocal free energies

Abels

Terasawa

2020

Math Methods in App Sciences

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We consider a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions and prove existence of weak solutions for it. In contrast to earlier contributions, we study a model with a singular nonlocal free energy, which controls the Hα/2‐norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.

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The Navier-Stokes Equations

Cited by 273 publications

References 48 publications

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Maximal $L^p$-Regularity for Stochastic Evolution Equations

On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions

Weak solutions for a diffuse interface model for two‐phase flows of incompressible fluids with different densities and nonlocal free energies

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