Viscosity effect on the degenerate lake equations

Taki, Bilal Al

doi:10.1016/j.na.2016.09.017

Cited by 2 publications

(10 citation statements)

References 29 publications

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“…Proof. The proof of lemma 4.4 follows the same line as in proposition 4.1. in [2]. For completeness sake, we will provide the proof again.…”

Section: Thus By Duality We Getmentioning

confidence: 96%

“…Here we just want to prove that Au ∈ L 2 (0, T ; V b ) and Bu ∈ L 2 (0, T ; V b ). The other properties are proved in [2]. Indeed, For u ∈ L ∞ (0, T ; H b ) ∩ L 2 (0, T ; V b ) and v ∈ V b , we estimate using Hölder's inequality…”

Section: and The Following Variational Inequality Holds For Allmentioning

confidence: 99%

“…For more details about Muckenhoupt classes and Weighted Sobolev spaces we refer the reader to [23] and [2].…”

Section: Bilal Al Taki Khawla Msheik and Jacques Sainte-mariementioning

confidence: 99%

“…Notice that in studying the Stokes problem, we just need to take q = 2 which is necessary to have a weighted version of Poincaré (or Korn's inequality). The reader can refer to lemma 4.1 in [2] for a detailed explanation of such choice.…”

Section: Bilal Al Taki Khawla Msheik and Jacques Sainte-mariementioning

confidence: 99%

“…Remark 3. Notice that, in [Proposition 3.1, [2]] , the first author proved a weighted version of Korn's inequality. However, some constraints on the weight b and the domain Ω were needed.…”

Section: Thus By Duality We Getmentioning

confidence: 99%

See 4 more Smart Citations

On the rigid-lid approximation of shallow water Bingham

Taki¹,

Msheik

Sainte-Marie³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. Duvaut and J.-L. Lions, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

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“…Proof. The proof of lemma 4.4 follows the same line as in proposition 4.1. in [2]. For completeness sake, we will provide the proof again.…”

Section: Thus By Duality We Getmentioning

confidence: 96%

Section: and The Following Variational Inequality Holds For Allmentioning

confidence: 99%

“…For more details about Muckenhoupt classes and Weighted Sobolev spaces we refer the reader to [23] and [2].…”

Section: Bilal Al Taki Khawla Msheik and Jacques Sainte-mariementioning

confidence: 99%

Section: Bilal Al Taki Khawla Msheik and Jacques Sainte-mariementioning

confidence: 99%

Section: Thus By Duality We Getmentioning

confidence: 99%

See 3 more Smart Citations

On the rigid-lid approximation of shallow water Bingham

Taki¹,

Msheik

Sainte-Marie³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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Degenerate lake equations: classical solutions and vanishing viscosity limit

Taki

Lacave

2022

Nonlinearity

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The objective of this paper is twofold. First, we show the existence of global classical solutions to the degenerate inviscid lake equations. This result is achieved after revising the elliptic regularity for a degenerate equation on the associated stream-function, and adapting the method used for construction of classical solutions to the incompressible Euler equations. Second, we show that the weak solutions of the viscous lake equations converge to classical solutions of the inviscid lake equations when the viscosity coefficient goes to zero, which constitutes an important physical validation of these models. The later result is achieved by the use of energy method as in the proofs of Kato-type theorems. This method also allows us to expose a convergence rate.

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Viscosity effect on the degenerate lake equations

Cited by 2 publications

References 29 publications

On the rigid-lid approximation of shallow water Bingham

On the rigid-lid approximation of shallow water Bingham

Degenerate lake equations: classical solutions and vanishing viscosity limit

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