Existence Theory for Stochastic Power Law Fluids

Breit, Dominic

doi:10.1007/s00021-015-0203-z

Cited by 26 publications

(34 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The p-Euler equations. In this subsection, we discuss the general properties of system (22) in more detail. To be convenient, we write out the system of equations below for x ∈ Ω t :…”

Section: 2mentioning

confidence: 99%

“…Here, ν > 0 and γ > 1 are constants. Note that the name 'p-Navier-Stokes equations' is reminiscent of the models for non-Newtonian fluids studied by Breit in [22,21] based on a power law model for the viscosity term (see Remark 4 for more details) 1 . The p-NS equations have scaling invariance (see Section 6.1.2) and therefore may admit self similar solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 4. The name 'p-Navier-Stokes equations' is reminiscent of the models for non-Newtonian fluids studied by Breit in [22,21]. We call (60) the 'p-Navier-Stokes equations' due to the Wasserstein-p distance behind.…”

mentioning

confidence: 99%

See 2 more Smart Citations

p-Euler equations and p-Navier–Stokes equations

Liu

2018

Journal of Differential Equations

View full text Add to dashboard Cite

Abstract. We propose in this work new systems of equations which we call p-Euler equations and p-Navier-Stokes equations. p-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-p distances, with incompressibility constraint. p-Euler equations have similar structures with the usual Euler equations but the 'momentum' is the signed (p − 1)-th power of the velocity. In the 2D case, the p-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the p-Laplacian of the streamfunction. By adding diffusion presented by γ-Laplacian of the velocity, we obtain what we call p-Navier-Stokes equations. If γ = p, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the p-Navier-Stokes equations in R d for γ = p and p ≥ d ≥ 2 through a compactness criterion.

show abstract

“…The p-Euler equations. In this subsection, we discuss the general properties of system (22) in more detail. To be convenient, we write out the system of equations below for x ∈ Ω t :…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

p-Euler equations and p-Navier–Stokes equations

Liu

2018

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…Much less is known if other fluid types are concerned. Just very recently, an analysis of non-Newtonian fluids (see [4,34,37]) and compressible fluids (see [6] and [32]) subject to stochastic forcing started. The analysis the system (1.5)-(1.6) brings a completely new aspect into play: a random variable exponent.…”

Section: Introductionmentioning

confidence: 99%

“…Ad (II) 4 . After we shall have passed to the supremum in the overall inequality, by Young's inequality we obtain for a finite constant C δ > 0…”

mentioning

confidence: 99%

Electro-rheological fluids under random influences: martingale and strong solutions

Breit

Gmeineder

2019

Stoch PDE: Anal Comp

Self Cite

View full text Add to dashboard Cite

We study generalised Navier-Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent p = p(ω, t, x) (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies p ≥ p − > 3n n+2 (p − > 1 in two dimensions). Under additional assumptions we obtain also stochastically strong solutions.

show abstract