Smoluchowski Navier-Stokes systems

Constantin, Peter

doi:10.1090/conm/429/08232

Cited by 14 publications

(19 citation statements)

References 20 publications

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“…The presence of the nonlinearity shows that the problem of deriving this equation from some underlying stochastic system is not trivial. The equation (5) has E as a Lyapunov functional. The time derivative of this Lyapunov functional is non-positive, vanishes at solutions of the Onsager equation, and if M is connected, only there:…”

Section: Introductionmentioning

confidence: 99%

“…The global existence of smooth solutions for small data for Oldroyd B-type models was established in ( [19], [23]) and for FENE in ( [27]). Global existence of smooth solutions for large data in 2D was established for Smoluchowski equations on compact manifolds ( [5], [7], [9], [10], [11], [30]). Global regularity for large data in the FENE case, under the corotational assumption was proved in ( [21], [26]).…”

Section: Introductionmentioning

confidence: 99%

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Note on Global Regularity for Two-Dimensional Oldroyd-B Fluids with Diffusive Stress

Constantin

Kliegl

2012

Arch Rational Mech Anal

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110

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Note on Global Regularity for Two-Dimensional Oldroyd-B Fluids with Diffusive Stress

Constantin

Kliegl

2012

Arch Rational Mech Anal

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110

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“…These two estimates make it possible, from (30) and as soon as d 2q + 1 r < 1 2 , to obtain (see [13,14] for similar results):…”

Section: Proof Of the Global Existence Resultsmentioning

confidence: 67%

Global Strong Solutions for Some Differential Viscoelastic Models

Chupin¹

2018

SIAM J. Appl. Math.

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The purpose of this article is to show that there are many differential viscoelastic models for which the global existence of a regular solution is possible. Although the problem of global existence in the classic Oldroyd model is still open, we show that by adding a non-linear contribution (proposed by R.G. Larson in 1984), it is possible to obtain more regular and global solutions, regardless of the size of the data (in the two-dimensional and periodic case). Similarly, more complex appearance models such as those related to "pom-pom" polymers are interesting and mathematically richer: some "natural" bounds on the stress make it possible to obtain global results. On the other hand, in the last part, we show that other models clearly do not seem to fit into this framework, and do not even seem to have a global solution in time. These kinds of results allow to highlight the advantages and disadvantages of such or such viscoelastic fluid models. They can thus help rheologists and numericists to make choices with new arguments.

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“…When the microscopic insertions have a larger number of degrees of freedom, m ∈ M , where M is a Riemannian manifold representing a finite number of degrees of freedom with constrains, then the kinetic equation is a Smoluchowski equation on the manifold. The upscaling principle advocated by one of us ( [5], [6]) is easiest formulated as the requirement that the sum of the energy of the fluid and the free energy of the microscopic quantities be a Lyapunov functional for the coupled system. In the known examples, this requirement leads to familiar rules of determining the added polymeric stress from the micro-micro and the macro-micro interactions ( [16]).…”

Section: Introductionmentioning

confidence: 99%

“…The global existence of smooth solutions for small data for Oldroyd B-type models was established in ( [17], [21]). Global existence of smooth solutions for large data in 2D was established for Smoluchowski equations on compact manifolds ( [6], [7], [9], [10], [11], [27]). Global regularity for large data in the FENE case, under the corotational assumption was proved in ( [19], [24]).…”

Section: Introductionmentioning

confidence: 99%

Remarks on Oldroyd-B and related complex fluid models

Constantin

Sun

2012

Communications in Mathematical Sciences

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Abstract. We prove global existence and uniqueness of solutions of Oldroyd-B systems, with relatively small data in R d , in a large functional setting (C α ∩ L 1 ). This is a stability result; solutions select an equilibrium and converge exponentially to it. Large spatial derivatives of the initial density and stress are allowed, provided the L ∞ norm of the density and stress are small enough. We prove global regularity for large data for a model in which the potential responds to high rates of strain in the fluid. We also prove global existence for a class of large data for a didactic scalar model which attempts to capture, in the simplest way, the essence of the dissipative nature of the coupling to fluid. This latter model has an unexpected cone invariance in function space that is crucial for the global existence.

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Smoluchowski Navier-Stokes systems

Abstract: We discuss equilibria, dynamics and regularity for Smoluchowski equations coupled to Navier-Stokes equations.

Cited by 14 publications

References 20 publications

Note on Global Regularity for Two-Dimensional Oldroyd-B Fluids with Diffusive Stress

Note on Global Regularity for Two-Dimensional Oldroyd-B Fluids with Diffusive Stress

Global Strong Solutions for Some Differential Viscoelastic Models

Remarks on Oldroyd-B and related complex fluid models

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