Liouville theorems for stationary flows of shear thickening fluids in 2D

Guo, Zherui

doi:10.5186/aasfm.2015.4052

Cited by 12 publications

(13 citation statements)

References 19 publications

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“…Bildhauer-Fuchs-Zhong [6] proved the weak solution is C 1,α by assuming h(t) = t 2 (1 + t) m , see also recent improved result by Jin-Kang in [19]. The Liouville property of (1.1) was started by Fuchs in [11], and later studied by Zhang in [25] and [26], where they obtained the trivial property of the solution with the help of u ∈ L ∞ or Ω h(|∇u|) < ∞. The degenerate case h(t) = t p was also considered by Bildhauer-Fuchs-Zhang in [5] by assuming that Ω |∇u| p < ∞.…”

Section: Introductionmentioning

confidence: 93%

Asymptotic properties of the plane shear thickening fluids with bounded energy integral

Li,

Wang,

Wang

2019

Preprint

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In this note we investigate the asymptotic behavior of plane shear thickening fluids around a bounded obstacle. Different from the Navier-Stokes case considered by Gilbarg-Weinberger in [18], where the good structure of the vorticity can be exploited and weighted energy estimates can be applied, we have to overcome the nonlinear term of high order. The decay estimates of the velocity was obtained by combining Point-wise Behavior Theorem in [16] and Brezis-Gallouet inequality in [7] together, which is independent of interest.

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Section: Introductionmentioning

confidence: 93%

Asymptotic properties of the plane shear thickening fluids with bounded energy integral

Li,

Wang,

Wang

2019

Preprint

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“…Let us note that during our calculations c is a generic constant independent of R and x 0 . Combining the above estimates with (22) and returning to (21), we get…”

Section: Note That Actually Limmentioning

confidence: 99%

“…We wish to remark that the proofs of the above results use the linearity of the leading part of (NSE) in an essential way. However, referring to the results obtained in [2], [3], [7], [21] and [22], we could show by applying appropriate arguments:…”

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confidence: 99%

A Liouville Theorem for Stationary Incompressible Fluids of Von Mises Type

Fuchs

Müller

2019

Acta Math Sci

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We consider entire solutions u of the equations describing the stationary flow of a generalized Newtonian fluid in 2D concentrating on the question, if a Liouville-type result holds in the sense that the boundedness of u implies its constancy. A positive answer is true for p-fluids in the case p > 1 (including the classical Navier-Stokes system for the choice p = 2), and recently we established this Liouville property for the Prandtl-Eyring fluid model, for which the dissipative potential has nearly linear growth. Here we finally discuss the case of perfectly plastic fluids whose flow is governed by a von Mises-type stress-strain relation formally corresponding to the case p = 1. it turns out that, for dissipative potentials of linear growth, the condition of µ-ellipticity with exponent µ < 2 is sufficient for proving the Liouville theorem.

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“…The validity of Liouville theorems has been established in the 2-D-case, i.e. for n = 2, for instance in the papers [3], [8], [9], [10], [11], [15], [21], [23], [29] and [30]. We like to mention that the case of potentials F satisfying (1.2) and (1.3) is treated in [10] assuming µ < 2.…”

Section: Introductionmentioning

confidence: 99%

Liouville-type results in two dimensions for stationary points of functionals with linear growth

Bildhauer

Fuchs

2022

Ann. Fenn. Math.

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We consider elliptic systems generated by variational integrals of linear growth satisfying the condition of \(\mu\)-ellipticity for some exponent \(\mu >1\) and prove that stationary points \(u\colon\mathbb{R}^2\to\mathbb{R}^N\) with the property \(\limsup_{|x|\to \infty} \frac{|u(x)|}{|x|} < \infty\) must be affine functions. The latter condition can be dropped in the scalar case together with appropriate assumptions on the energy density providing an extension of Bernstein's theorem.

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Liouville theorems for stationary flows of shear thickening fluids in 2D

Abstract: Abstract. In this paper we consider entire weak solutions u of the equations for stationary flows of shear thickening fluids in the plane and prove Liouville theorems in case of finite energy or under suitable integrability assumptions on u.

Cited by 12 publications

References 19 publications

Asymptotic properties of the plane shear thickening fluids with bounded energy integral

Asymptotic properties of the plane shear thickening fluids with bounded energy integral

A Liouville Theorem for Stationary Incompressible Fluids of Von Mises Type

Liouville-type results in two dimensions for stationary points of functionals with linear growth

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