Mathematical Theory of Incompressible Nonviscous Fluids

Marchioro, Carlo; Pulvirenti, Mario

doi:10.1007/978-1-4612-4284-0

Cited by 664 publications

(766 citation statements)

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“…The reader interested in such issues may wish to consult the book by Marchioro and Pulvirenti. 35 Our viewpoint in this article will be to treat Eqs. ͑1͒ or ͑2͒ as an interesting dynamical system in its own right and proceed from there.…”

Section: The Basic Dynamical Equationsmentioning

confidence: 99%

Point vortex dynamics: A classical mathematics playground

Aref¹

2007

Journal of Mathematical Physics

199

213

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The idealization of a two-dimensional, ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics. Several examples are given including the integrability of the three-vortex problem, the interplay of relative equilibria of identical vortices and the roots of certain polynomials, addition formulas for the cotangent and the Weierstraß function, projective geometry, and other topics. The hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community.

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Section: The Basic Dynamical Equationsmentioning

confidence: 99%

Point vortex dynamics: A classical mathematics playground

Aref¹

2007

Journal of Mathematical Physics

199

213

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“…This approach allows an easier justification of the different approximations needed to obtain the mean thermodynamic vorticity equation. One condition for the validity of the thermodynamical approach is that the system of point vortices be "mixing" [25], [45]. In fact, as our goal was to apply thermodynamics to a subsystem, that is in some localized region and for a limited period of time, the assumption of mixing, and even of the existence of the thermodynamic limit is less restrictive than in the general case of the global relaxation of Euler flows because of the presence of energy fluctuations.…”

Section: Discussionmentioning

confidence: 99%

“…(A rigorous derivation of the motion equations of point vortices, using the distribution formalism, may be found in Marchioro & Pulvirenti [25].) These point vortices were first introduced by Stewart [42], in his study of vortical atmospheric motion in the geostrophic approximation.…”

Section: Thermodynamic Formalism and Mean Field Vorticity Equationmentioning

confidence: 99%

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Relaxation towards localized vorticity states in drift plasma and geostrophic flows

Agullo¹,

Verga

2004

Phys. Rev. E

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The drift of ions in a magnetized plasma or the height fluctuations of a rotating fluid layer are described by the conservation equation of a potential vorticity. This potential vorticity contains an intrinsic length scale, the hybrid Larmor radius in plasma, and the Rossby length in the quasigeostrophic flow. The influence of this scale in the evolution of a random initial vorticity field is investigated using a thermodynamic approach. In contrast to the perfect fluid case, where the vorticity tends to a well defined stationary state, complete relaxation towards an equilibrium state is not observed in numerical simulations of quasi-geostrophic decaying turbulence. The absence of global thermodynamic equilibrium is explained by the relaxation towards states of local equilibrium where the vorticity is concentrated. Interaction between these separated regions is extremely weak. Explicit, axisymmetric, localized solutions of the mean field integro-differential equation of extremal entropy states are obtained using asymptotic methods. Comparison of the computed solutions with the observed coherent structures, shows that they effectively correspond to states in local thermodynamic equilibrium.

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“…We note that for the 2D Euler equation, the definition of weak solutions via (1.7) can be found, for instance, in [21,Theorem 3.2], and that it easily implieŝ…”

Section: Remarks 1 the Last Claim Means That Eithermentioning

confidence: 99%

Local Regularity for the Modified SQG Patch Equation

Kiselev

Yao

Zlatoš

2016

Comm Pure Appl Math

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We study the patch dynamics on the whole plane and on the half-plane for a family of active scalars called modified SQG equations. These involve a parameter α which appears in the power of the kernel in their Biot-Savart laws and describes the degree of regularity of the equation. The values α = 0 and α = 1 2 correspond to the 2D Euler and SQG equations, respectively. We establish here local-in-time regularity for these models, for all α ∈ (0, 1 2 ) on the whole plane and for all small α > 0 on the half-plane. We use the latter result in [16], where we show existence of regular initial data on the half-plane which lead to a finite time singularity.

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Mathematical Theory of Incompressible Nonviscous Fluids

Cited by 664 publications

References 0 publications

Point vortex dynamics: A classical mathematics playground

Point vortex dynamics: A classical mathematics playground

Relaxation towards localized vorticity states in drift plasma and geostrophic flows

Local Regularity for the Modified SQG Patch Equation

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