Regularity theorems, up to the boundary, for shear thickening flows

Veiga, H. Beirão da; Kaplický, Petr; Růžička, Michael

doi:10.1016/j.crma.2010.04.010

Cited by 9 publications

(9 citation statements)

References 11 publications

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“…Since the framework of implicitly constituted fluids characterized by (A1)-(A5) is more general than the setting considered in previous studies, the result established in Theorem 1.1 provides large-data existence theory to a broader class of models in comparison with earlier studies (we refer the reader to the survey paper [43] and the recent studies [15,19] for detailed summaries on long-time and large-data analysis of power-law-type models). In particular, it follows from Theorem 1.1 and Lemma 1.1 that (for large-data) there is weak solution to Bingham or Herschel-Bulkley fluids (1.28) (or (1.14) with ν(s) ∼ s r−2 ) if r > 6 5 in three spatial dimensions-the result that is not covered by any of the previous studies. 9 The class of fluids, to which the result is applicable, is, however, much larger, as indicated in subsection 1.1.…”

Section: Main Resultmentioning

confidence: 89%

“…See Schwedoff[54], Troutan[61], and further references in books on non-Newtonian fluids, such as Bird, Amstrong, and Hassager[9], Huilgol[31], and Schowalter[53], or in the survey paper[44].2 Theoretical analysis initiated by Ladyzhenskaya[35,36] (see also Lions[37]) has developed extensively during the last few decades; see, for example, studies of different types[6,7,10,19,18,28,38,39,40,62].…”

mentioning

confidence: 99%

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On Unsteady Flows of Implicitly Constituted Incompressible Fluids

Buli¹,

Gwiazda²,

Alek³

et al. 2012

SIAM J. Math. Anal.

116

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We consider unsteady flows of incompressible fluids with a general implicit constitutive equation relating the deviatoric part of the Cauchy stress S and the symmetric part of the velocity gradient D in such a way that it leads to a maximal monotone (possibly multivalued) graph and the rate of dissipation is characterized by the sum of a Young function depending on D and its conjugate being a function of S. Such a framework is very robust and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or Herschel-Bulkley type, and shear rate-dependent fluids with discontinuous viscosities as special cases. The appearance of S and D in all the assumptions characterizing the implicit relationship G(D, S) = 0 is fully symmetric. We establish long-time and large-data existence of weak solution to such a system completed by the initial and the Navier slip boundary conditions in both the subcritical and supercritical cases. We use tools such as Orlicz functions, properties of spatially dependent maximal monotone operators, and Lipschitz approximations of Bochner functions taking values in Orlicz-Sobolev spaces.

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Section: Main Resultmentioning

confidence: 89%

mentioning

confidence: 99%

On Unsteady Flows of Implicitly Constituted Incompressible Fluids

Buli¹,

Gwiazda²,

Alek³

et al. 2012

SIAM J. Math. Anal.

116

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“…The author shows that the second "tangential" derivatives belong to L 2 (R 3 + ), while the second "normal" derivatives belong to some L l loc (R 3 + ), for a suitable l < 2. See [9] and [10] for recent, and more general, related results (under the non-slip boundary condition), and for references.…”

Section: Independence Of the Compatibility Conditionsmentioning

confidence: 99%

Reducing slip boundary value problems from the half to the whole space. Applications to inviscid limits and to non-Newtonian fluids

Veiga

Crispo

Grisanti

2011

Journal of Mathematical Analysis and Applications

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The study of a very large class of linear and non-linear, stationary and evolutive partial differential problems in the half-space (or similar) under the slip boundary condition is reduced here to the much simpler study of the corresponding results for the same problem in the whole space. The approach is particularly suitable for proving new results in strong norms. To determine whether this extension is available, turns out to be a simple exercise. The verification depends on a few general features of the functional space X related to the space variables. Hence, we present an approach as much as possible independent of the particular space X. We appeal to a reflection technique. Hence a crucial assumption is to be in the presence of flat boundaries (see below). Instead of stating "general theorems" we rather prefer to illustrate how to apply our results by considering a couple of interesting problems. As a main example, we show that sharp vanishing viscosity limit results that hold for the evolution Navier-Stokes equations in the whole space can be extended to the slip boundary value problem in the half-space. We also show some applications to non-Newtonian fluid problems

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“…had generalized these results to p > 8 5 in [23] and p > 6 5 in [24]. There are also many papers dealing with regularity of for evolution Dirichlet boundary problems and we refer instance to [3,4,16,8,9,10,11]. For the study of these models with space periodic boundary conditions, we refer to the monograph [34] and papers [14,22].…”

Section: Introductionmentioning

confidence: 96%

“…The results of regularity up to boundary for the Dirichlet problems were obtained by T. Shinlkin [41] and H. Beirão da Veiga, et.al. [5,6,7,8,9,10,11,20,21].…”

Section: Introductionmentioning

confidence: 99%

Existence of weak solutions for non-stationary flows of fluids with shear thinning dependent viscosities under slip boundary conditions in half space

Zang

2017

Sci. China Math.

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The author treats the system of motion for an incompressible non-Newtonian fluids of the stress tensor described by p−potential function subject to slip boundary conditions in R 3 + . Making use of the Oseentype approximation to this model and the L ∞ -truncation method, one can establish the existence theorem of weak solutions for p−potential flow with p ∈ ( 8 5 , 2] provided that large initial data.

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Regularity theorems, up to the boundary, for shear thickening flows

Cited by 9 publications

References 11 publications

On Unsteady Flows of Implicitly Constituted Incompressible Fluids

On Unsteady Flows of Implicitly Constituted Incompressible Fluids

Reducing slip boundary value problems from the half to the whole space. Applications to inviscid limits and to non-Newtonian fluids

Existence of weak solutions for non-stationary flows of fluids with shear thinning dependent viscosities under slip boundary conditions in half space

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