Continuity of weak solutions of the critical surface quasigeostrophic equation on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>

Alonso-Orán, Diego; Córdoba, Antonio; Martínez, Ángel D.

doi:10.1016/j.aim.2018.01.015

Cited by 5 publications

(7 citation statements)

References 26 publications

(38 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In passing, we remark that despite the lack of literature on the analysis of Boussinesq equations on manifolds, various PDEs of hydrodynamic models have nevertheless been studied on manifolds, including the Navier-Stokes equations [40], the rotating Euler equations [55], and the SQG (surface quasi-geostrophic) equations [6,7], even for critical cases without the smallness assumption on the initial data.…”

Section: Siran LI Jiahong Wu and Kun Zhaomentioning

confidence: 99%

See 1 more Smart Citation

On the degenerate boussinesq equations on surfaces

Li¹,

Wu²,

Zhao³

2020

Journal of Geometric Mechanics

View full text Add to dashboard Cite

In this paper we study the non-degenerate and partially degenerate Boussinesq equations on a closed surface Σ. When Σ has intrinsic curvature of finite Lipschitz norm, we prove the existence of global strong solutions to the Cauchy problem of the Boussinesq equations with full or partial dissipations. The issues of uniqueness and singular limits (vanishing viscosity/vanishing thermal diffusivity) are also addressed. In addition, we establish a breakdown criterion for the strong solutions for the case of zero viscosity and zero thermal diffusivity. These appear to be among the first results for Boussinesq systems on Riemannian manifolds.

show abstract

Section: Siran LI Jiahong Wu and Kun Zhaomentioning

confidence: 99%

“…Again, in view of the breakdown criterion (Theorem 3.4), the Sobolev-Morrey embedding W 2,p (Σ) → W 1,∞ (Σ) for p > 2 and the uniqueness Theorems 3.2 and 3.3, the proof is now complete. 6. Vanishing viscosity and diffusivity limits.…”

mentioning

confidence: 99%

On the degenerate boussinesq equations on surfaces

Li¹,

Wu²,

Zhao³

2020

Journal of Geometric Mechanics

View full text Add to dashboard Cite

show abstract

“…Corollary 1.5. Let (M, g) be a compact riemannian manifold of dimension n, s ∈ (0, 1 2 ) and p = 2n n−2s . Then there exist a constant C > 0 such that…”

Section: Introductionmentioning

confidence: 99%

Integral representation for fractional Laplace–Beltrami operators

Alonso-Orán

Córdoba

Martínez

2018

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.

show abstract

“…One studies the evolution of some class of initial data. In their previous work [2] the authors established an explicit modulus of continuity for weak solutions: Theorem 1.1. Given an initial datum θ 0 ∈ L 2 (S 2 ) any weak solution of (1.1) becomes instantanously continuous with an explicit modulus of continuity, for any time t > 0.…”

Section: Introductionmentioning

confidence: 99%

“…Global well-posedness of the critical surface quasigeostrophic equation on the sphere. The critical surface quasigeostrophic equation on the sphere has been treated in [2] and is given by…”

Section: Introductionmentioning

confidence: 99%