A Maximum Principle Applied to Quasi-Geostrophic Equations

Abstract. In this paper we study the analytic structure of a two-dimensional mass balance equation of an incompressible fluid in a porous medium given by Darcy's law. We obtain local existence and uniqueness by the particle-trajectory method and we present different global existence criterions. These analytical results with numerical simulations are used to indicate nonformation of singularities. Moreover, blow-up results are shown in a class of solutions with infinite energy.

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“…First, we calculate the derivative of m as in [5,9]. Let s be a point of differentiability of m(t), then for τ > 0…”

Section: Singularities With Infinite Energymentioning

confidence: 99%

Analytical behavior of two-dimensional incompressible flow in porous media

Córdoba

Gancedo

Orive

2007

Journal of Mathematical Physics

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“…In contrast with the proof in [CC04] which relies on explicit integral representations of the fractional Laplacian for all the arguments, the present proof includes hypothesis that are functional analysis (∆ generates a contraction semigroup in L p ) and geometric (e t∆ satisfies a comparsison principle). Hence they apply just as well to the cases in which ∆ is a second order uniformly elliptic operator in divergence form.…”

Section: Introductionmentioning

confidence: 99%

“…In many applications, the vector v depends on the vector u. For the quasigeostrophic equation in dimension 2 (see [CC04] and references therein) u = u(x, t) has the physical meaning of a potential temperature, v = v(x, t) is the velocity vector, and κ 0 is the viscosity. In the physical framework, u and v are related by the equation v = −∇ ⊥ (−∆) −1/2 u.…”

Section: Introductionmentioning

confidence: 99%

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L p -bounds for quasi-geostrophic equations via functional analysis

Llave

Valdinoci

2011

Journal of Mathematical Physics

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“…29) allow us to accomplish the fact that m (t) = σ t (α t , t) for almost all t. Since we can use Eqs. 13 and 17 and the above estimates, it is possible to control σ t (α t , t) L ∞ by means of E RT (t).…”

Section: The Evolution Of the Rayleigh-taylor Conditionmentioning

confidence: 99%

The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces

Córdoba

Córdoba²,

Gancedo³

2009

Proc. Natl. Acad. Sci. U.S.A.

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For the free boundary dynamics of the two-phase Hele-Shaw and Muskat problems, and also for the irrotational incompressible Euler equation, we prove existence locally in time when the RayleighTaylor condition is initially satisfied for a 2D interface. The result for water waves was first obtained by Wu in a slightly different scenario (vanishing at infinity), but our approach is different because it emphasizes the active scalar character of the system and does not require the presence of gravity.Euler | Hele-Shaw-Muskat | incompressible | well-possedness T here are several interesting problems in fluid mechanics regarding the evolution of the interface between two fluids [the Hele-Shaw cell (1, 2) and the Muskat problem (3)] or between a fluid and vacuum or another fluid with zero density, as in models of water waves. In all of them the first important question to be asked is to guarantee local existence, usually within the chain of Sobolev spaces. However, such a result turns out to be false for general initial data. First, Rayleigh (4), Taylor (5), and Saffman and Taylor (2), and later Beale et al. (6), Wu (7,8), Christodoulou and Lindblad (9), Ambrose (10), Lindblad (11), Ambrose and Masmoudi (12), Coutand and Shkoller (13), , Shatah and Zeng (15) and Zhang and Zhang (16) gave a condition that must be satisfied to have a solution locally in time; namely, the normal component of the pressure gradient jump at the interface has to have a distinguished sign. This is known as the Rayleigh-Taylor condition.In refs. 17 and 18 we have obtained local existence in the 2D case: for the Hele-Shaw and Muskat problems, our result addresses the more difficult case when the two fluids have different densities and viscosities; for water waves, we give a different proof of the important theorem of Wu (7) where gravity plays a crucial role in the sign of the Rayleigh-Taylor condition. In our proof, however, we consider the two cases, with or without gravity, and with initial data always satisfying the Rayleigh-Taylor condition.When that condition is not imposed initially there are several cases where ill-posedness has been proved. Let us point out the works of Ebin (19,20), Caflisch and Orellana (21), Siegel et al. (22), and Córdoba and Gancedo (14).We regard these models as transport equations for the density, considered as an active scalar, with a divergence-free velocity field given by Darcy's law (Hele-Shaw and Muskat) or Bernoulli's law (irrotational incompressible Euler equation). It follows that the vorticity is then a delta distribution at the interface multiplied by an amplitude. The dynamics of that interface is governed by the Birkhoff-Rott integral of the amplitude from which we may subtract any component in the tangential direction without modifying its evolution (see ref. 23). We treat the case without surface tension which leads to equality of the pressure on the free boundary, and in both problems it is assumed that the initial interface does not self-intersect. We quantify that property by imposing that the a...

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A Maximum Principle Applied to Quasi-Geostrophic Equations

Cited by 540 publications

References 17 publications

Analytical behavior of two-dimensional incompressible flow in porous media

Analytical behavior of two-dimensional incompressible flow in porous media

L p -bounds for quasi-geostrophic equations via functional analysis

The Rayleigh-Taylor condition for the evolution of irrotational fluid interfaces

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