Estimates for solutions of the nonstationary stokes problem in anisotropic Sobolev spaces with mixed norm

Maremonti, Paolo; Solonnikov, V. A.

doi:10.1007/bf02355828

Cited by 32 publications

(37 citation statements)

References 9 publications

(19 reference statements)

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“…For this reason they were taken in [5] to prove that functions |∂ t v|, |∇ 2 v| and |∇p| belong to L 5 4 (Q T ). In [2], [11] the theorem on unique solvability of the first initial boundary value problem for the Stokes system was extended to the case of spaces W 2,1 s,l (Q T ) × W 1,0 s,l (Q T ) with two different exponents s, l ∈]1, ∞[. Using this theorem, bound (1.4) and arguing as in [5], one can make the following conclusion.…”

Section: Inroductionmentioning

confidence: 77%

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations

Ladyzhenskaya

Seregin

1999

Journal of Mathematical Fluid Mechanics

249

237

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We prove a criterion of local Hölder continuity for suitable weak solutions to the Navier-Stokes equations. One of the main part of the proof, based on a blow-up procedure, has quite general nature and can be applied to other problems in spaces of solenoidal vector fields. Mathematics Subject Classification (1991). 35K, 76D.In the present paper we deal with weak solutions to the three-dimensional NavierStokes equationsHopf proved the global existence at least one weak solution v to the first initial boundary value problem with boundary condition v| ∂Ω×[0,T ] = 0 under quite general assumptions on domain Ω, external force f and initial data v| t=0 = a. In [5] his results were discribed and the class of Hopf's solutions was introduced. Corresponding definition includes all main properties that essentially were proved by E. Hopf. More precisely, a velocity field v is called Hopf's solution if it belongs to T ] in the weak topology of L 2 (Ω; R 3 ) and satisfies the integral identityfor all t ∈ [0, T ] and for all w ∈ • J 1 2 (Ω). No information on the pressure p is given. Here • J(Ω) is the L 2 (Ω; R 3 )-closure of the set of all smooth solenoidal fields

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Section: Inroductionmentioning

confidence: 77%

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations

Ladyzhenskaya

Seregin

1999

Journal of Mathematical Fluid Mechanics

249

237

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“…By repeating the above argument with v in place of v x 1 , we complete the proof of the inequality (18). The lemma is proved.…”

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confidence: 75%

“…There has been, in fact, considerable study of parabolic equations in mixed norm spaces in the literature (see, e.g., [4,18,11,10,12,21,3,6,20,14] and references therein). The usual mixed norms are of the form L q ((0, T ), L p ), that is, q summability in the time variable and p summability in the spatial variables.…”

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confidence: 99%

Elliptic and Parabolic Equations with Measurable Coefficients in $L_p$-spaces with Mixed Norms

Kim¹

2008

Methods and Applications of Analysis

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Abstract. The unique solvability results for second order parabolic and elliptic equations in Sobolev spaces with mixed norms are presented. The second order coefficients are measurable in one spatial variable and VMO (vanishing mean oscillation) in the other spatial variables. In the parabolic case, the coefficients (except a 11 ) are further allowed to be only measurable in time. We first prove the solvability results for equations in the whole Euclidean space. Then, using these results as well as some extension techniques, we prove the solvability results for equations on a half space without any boundary estimates. The mixed norms we present here are more general than the usual mixed norm L t q L x p .Key words. Second order elliptic and parabolic equations, vanishing mean oscillation, Sobolev spaces with mixed norms.

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“…First observe that under the hypothesis of Theorem 1.7, we have u(·, t) L 4 (R 3 ) ≤ C u(·, t) where the latter follows from Hölder's inequality. Using these inclusions, the coercive estimates (see [9,22,36]) and the uniqueness theorem (see [16]) for the Stokes problem, we can introduce the associated pressure p such that…”

Section: Proof Of Theorems 17mentioning

confidence: 99%

The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

Phuc

2015

J. Math. Fluid Mech.

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Estimates for solutions of the nonstationary stokes problem in anisotropic Sobolev spaces with mixed norm

Cited by 32 publications

References 9 publications

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations

Elliptic and Parabolic Equations with Measurable Coefficients in $L_p$-spaces with Mixed Norms

The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

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Estimates for solutions of the nonstationary stokes problem in anisotropic Sobolev spaces with mixed norm

Cited by 32 publications

References 9 publications

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations

Elliptic and Parabolic Equations with Measurable Coefficients in $L_p$-spaces with Mixed Norms

The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

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On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations

On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations