Results 1  10
of
194
POSITIVE SOLUTIONS OF NONLINEAR PROBLEMS INVOLVING THE SQUARE ROOT OF THE LAPLACIAN
"... Abstract. We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a pri ..."
Abstract

Cited by 82 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of GidasSpruck type. In addition, among other results, we prove a symmetry theorem of GidasNiNirenberg type. 1.
On the Euler equations of incompressible fluids
 Bull.Amer. Math. Soc
, 2007
"... Abstract. Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of illposed free surface problems such as ..."
Abstract

Cited by 62 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of illposed free surface problems such as the RayleighTaylor and KelvinHelmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned. 1.
Regularity of Hölder continuous solutions of the supercritical quasigeostrophic equation (2007), available at arxiv:math.AP/0701592
"... Abstract. We present a regularity result for weak solutions of the 2D quasigeostrophic equation with supercritical (α < 1/2) dissipation (−∆)α: If a LerayHopf weak solution is Hölder continuous θ ∈ Cδ(R2) with δ> 1 − 2α on the time interval [t0, t], then it is actually a classical solution ..."
Abstract

Cited by 54 (12 self)
 Add to MetaCart
(Show Context)
Abstract. We present a regularity result for weak solutions of the 2D quasigeostrophic equation with supercritical (α < 1/2) dissipation (−∆)α: If a LerayHopf weak solution is Hölder continuous θ ∈ Cδ(R2) with δ> 1 − 2α on the time interval [t0, t], then it is actually a classical solution on (t0, t].
Potential theory of subordinate Brownian motions revisited’, Stochastic analysis and applications to finance–essays
 in honour of Jiaan Yan, (eds
, 2012
"... The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger ..."
Abstract

Cited by 43 (18 self)
 Add to MetaCart
(Show Context)
The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger conditions.
Global wellposedness and a decay estimate for the critical dissipative quasigeostrophic equation in the whole space. Discrete Contin
 Dyn. Syst
"... Abstract. We study the critical dissipative quasigeostrophic equations in R2 with arbitrary H1 initial data. After showing certain decay estimate, a global wellposedness result is proved by adapting the method in [10]. A decay in time estimate for higher Sobolev norms of solutions is also discusse ..."
Abstract

Cited by 36 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We study the critical dissipative quasigeostrophic equations in R2 with arbitrary H1 initial data. After showing certain decay estimate, a global wellposedness result is proved by adapting the method in [10]. A decay in time estimate for higher Sobolev norms of solutions is also discussed. 1.
Global regularity for a modified critical dissipative quasigeostrophic equation
 Indiana Univ. Math. J
, 2008
"... Abstract. In this paper, we consider the modified quasigeostrophic equation ∂tθ + (u · ∇) θ + κΛ α θ = 0 u = Λ α−1 R ⊥ θ. with κ> 0, α ∈ (0, 1] and θ0 ∈ L 2 (R 2). We remark that the extra Λ α−1 is introduced in order to make the scaling invariance of this system similar to the scaling invarianc ..."
Abstract

Cited by 33 (9 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we consider the modified quasigeostrophic equation ∂tθ + (u · ∇) θ + κΛ α θ = 0 u = Λ α−1 R ⊥ θ. with κ> 0, α ∈ (0, 1] and θ0 ∈ L 2 (R 2). We remark that the extra Λ α−1 is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasigeostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system. (1.1) (1.2)
ON THE GLOBAL WELLPOSEDNESS OF THE CRITICAL QUASIGEOSTROPHIC EQUATION
, 2007
"... Abstract. We prove the global wellposedness of the critical dissipative quasigeostrophic equation for large initial data belonging to the critical Besov space ˙ B 0 ∞,1(R 2). 1. ..."
Abstract

Cited by 33 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We prove the global wellposedness of the critical dissipative quasigeostrophic equation for large initial data belonging to the critical Besov space ˙ B 0 ∞,1(R 2). 1.
GLOBAL WELLPOSEDNESS FOR EULERBOUSSINESQ SYSTEM WITH CRITICAL DISSIPATION
, 903
"... Abstract. In this paper we study a fractional diffusion Boussinesq model which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion for the temperature. We prove global wellposedness results. Boussinesq systems of the type ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
Abstract. In this paper we study a fractional diffusion Boussinesq model which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion for the temperature. We prove global wellposedness results. Boussinesq systems of the type