The Maximum Principle and the Global Attractor for the Dissipative 2D Quasi-Geostrophic Equations

Ju, Ning

doi:10.1007/s00220-004-1256-7

Cited by 204 publications

(143 citation statements)

References 25 publications

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“…The following pointwise estimate for fractional derivatives is proved in [13] and represents an improvement of an earlier estimate by Córdoba and Cór-doba [11].…”

Section: Under All the Above Assumptions We Havementioning

confidence: 58%

Exploding solutions for a nonlocal quadratic evolution problem

Liu¹,

Rodrigo²,

Zhang³

2010

Rev. Mat. Iberoam.

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We consider a nonlinear parabolic equation with fractional diffusion which arises from modelling chemotaxis in bacteria. We prove the wellposedness, continuation criteria and smoothness of local solutions. In the repulsive case we prove global wellposedness in Sobolev spaces. Finally in the attractive case, we prove that for a class of smooth initial data the L ∞ x -norm of the corresponding solution blows up in finite time. This solves a problem left open by Biler and Woyczyński [8].

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“…The following pointwise estimate for fractional derivatives is proved in [13] and represents an improvement of an earlier estimate by Córdoba and Cór-doba [11].…”

Section: Under All the Above Assumptions We Havementioning

confidence: 58%

Exploding solutions for a nonlocal quadratic evolution problem

Liu¹,

Rodrigo²,

Zhang³

2010

Rev. Mat. Iberoam.

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“…For the proof of Lemma 2.1, see [4,5,10]. For k ∈ Z + , σ ≥ 0, 1 ≤ p ≤ ∞ and t > 0, the Poisson kernel fulfills…”

Section: Preliminariesmentioning

confidence: 99%

“…10) are written byP (t) * ω 0 = M ω P (t) + m ω ∂ x P (t) + ρ 0 (t − s) * (u∂ x u) (s)ds = M (t − s) * (P ∂ x P )(s)ds − ρ 4 (t) respectively, where ρ 0 (t) =P (t) * ω 0 − M ω P (t) − m ω ∂ x P (t − s) * ∂ x ω(s) 2 − M 2 ω P (s) x P (t − s) * (ω 2 )(x P (t − s) * (ω(s) 2 − M 2 ω P (1 + s) ∂ x P (t − s, x − y) − ∂ x P (t, x)) P (s, y) 2 dyds − ρ 1 (t) − ρ 2 (t) − ρ 3 x P (t − s, x − y) − ∂ x P (t, x)) × (ω(s, y) 2 − M 2 ω P (1 + s, y) s, y) 2 − M 2 ω P (1 + s, y) ∂ x P (t − s, x − y) − ∂ x P (t, x))× (P (1 + s, y) 2 − P (s, y) 2 )dyds.…”

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Asymptotic Behavior of Solutions to the Drift-Diffusion Equation with Critical Dissipation

Yamamoto

Sugiyama

2015

Ann. Henri Poincaré

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In this paper, the initial value problem for the drift-diffusion equation which stands for a model of a semiconductor device is studied. When the dissipative effect on the drift-diffusion equation is given by the half Laplacian, the dissipation balances to the extra force term. This case is called critical. The goal of this paper is to derive decay and asymptotic expansion of the solution to the drift-diffusion equation as time variable tends to infinity.

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“…This well-posedness problem was also examined by Córdoba and Córdoba [13] in H m spaces for α > 0. For α > 1 2 (the subcritical case), it is well known (see [11,18]) that a smooth initial function θ 0 with a compact support leads to a global regular solution. For the critical case α = 1 2 , similar to the three-dimensional Navier-Stokes equations, the nonlinear term is difficult to be controlled by the dissipative term for large solutions.…”

Section: Introductionmentioning

confidence: 99%

“…However, the global existence is obtainable if additional conditions are applied (see, for example, Constantin and Wu [12] and Dong and Pavlovic [15]). One may also refer to Chen and Xiong [8] for computational steady-state bifurcation results of Charney and DeVore quasi-geostrophic equation and to Kato and Ponce [22], Ju [18], Danchin [14] and Chen and Chen [3] for commutator estimates applicable to equation (1).…”

Section: Introductionmentioning

confidence: 99%