Hydrodynamics in Besov Spaces

Vishik, Mikhail M

doi:10.1007/s002050050128

Cited by 199 publications

(171 citation statements)

References 2 publications

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“…For this purpose, one may use again Bony's decomposition, the fact that div σ = 0 and classical continuity properties for the paraproduct and remainder operators. One ends up for instance with: (14) and Young inequality, it is now easy to get an inequality similar to (16), and thus a bound for θ in…”

Section: A Global Results For Infinite Energy Initial Velocitymentioning

confidence: 99%

“…Indeed, according to a result by M. Vishik in [16] concerning the transport equation, one can propagate the B 0 ∞,1 regularity over the vorticity ω provided ∂ 1 θ is in L 1 loc (R + ; B 0 ∞,1 ) and there exists some universal constant C such that (14), one may write…”

Section: Further Results and Concluding Remarksmentioning

confidence: 99%

“…There, global well-posedness is shown whenever the initial velocity u 0 belongs to B 1+ 2 p p,1 and the initial temperature θ 0 is in L r for some (p, r) satisfying 2 < r ≤ p ≤ ∞ (plus a technical condition if p = r = ∞). Let us emphasize that in the Besov spaces framework, the assumption on u 0 is somewhat optimal (since it is optimal for the standard Euler equations, see [16]). …”

Section: Introductionmentioning

confidence: 99%

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Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

Danchin

Paicu²

2009

Commun. Math. Phys.

174

102

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Abstract. The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. None smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the Bénard system are also discussed.

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Section: A Global Results For Infinite Energy Initial Velocitymentioning

confidence: 99%

Section: Further Results and Concluding Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

Danchin

Paicu²

2009

Commun. Math. Phys.

174

102

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“…The proof of the local existence for v 0 ∈ B 1 ∞,1 (R 3 ) is implied in the proofs of the main theorems in [5,6](see also [35]), and explicitly written in [27]. The above theorem implies that if T * is the first time of singularity, then we have the lower estimate of the blow-up rate,…”

Section: The Euler Equationsmentioning

confidence: 96%

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Chae

2007

Math. Ann.

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In this paper we rule out the possibility of asymptotically selfsimilar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T . For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm,Ḃ 0 1,∞ (R 3 ). For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in L q (B(z, r)) for some q ∈ [2, ∞), where the ball of radius r is shrinking toward a possible singularity point z at the order of √ T − t as t approaches to T . In the L q (R 3 ) convergence case with q ∈ [3, ∞) we present a simple alternative proof of the similar result in [16]. *

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“…We also mention the survey paper [3] on existence of spatially decaying solutions of the Navier-Stokes problem in various domains (not necessarily satisfying (0.4)); see also [11] and [26].…”

Section: Introduction It Is Well Known That the Navier-stokes Systemmentioning

confidence: 99%

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

Zelik¹

2007

Glasgow Math. J.

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Hydrodynamics in Besov Spaces

Cited by 199 publications

References 2 publications

Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Spatially Nondecaying Solutions of the 2d Navier-Stokes Equation in a Strip

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