On the “Mean Field” Interpretation of Burgers' Equation

Choquard, Ph.; Wagner, Joël

doi:10.1023/b:joss.0000037211.80229.04

Cited by 20 publications

(54 citation statements)

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“…This is accomplished without any supplementary conditions on S such as uniform continuity or bounds; see Theorem 2. We remark that this result is independent of the continuity equation (7) and that the hypothesis on S 0 is weaker than that which we will impose when adjoining (7) to (6). It is also shown that a suitable transformation of variables reduces the problem to the HamiltonJacobi equation for a free particle.…”

Section: Outline and Principal Resultsmentioning

confidence: 68%

“…Sections 3 through 6 are devoted to the viscosity-measure solution (S, ρ) of the initial-value problem (6,7).…”

Section: Outline and Principal Resultsmentioning

confidence: 99%

“…This means that we investigate viscosity solutions [8] of (6) and measure solutions [14] of (7). This combination of generalized solutions was recently extensively studied in [1].…”

mentioning

confidence: 99%

“…The Lagrangian function associated with (6) is L(ẋ, x) = 1 2 (ẋ 2 − ω 2 x 2 ). Notice that L does not fulfill the minorization condition L(ẋ, x) ≥ f (|ẋ|) with f (s)/s → ∞ as s → ∞ which is customarily assumed in the literature on the calculus of variations and viscosity solutions.…”

mentioning

confidence: 99%

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A One-Dimensional Inviscid and Compressible Fluid in a Harmonic Potential Well

Choquard

Strömberg

2007

Acta Appl Math

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A Hamiltonian model is analyzed for a one-dimensional inviscid compressible fluid. The space-time evolution of the fluid is governed by the following system of the Hamilton-Jacobi and the continuity equations:Here S and ρ designate the velocity potential and the mass density, respectively. Unless S 0 is convex, shocks form and the velocity S x becomes discontinuous in {0 < ωt < π/2}. It is demonstrated that there nevertheless exists a unique viscosity-measure solution (S, ρ) when S 0 is globally Lipschitz continuous and locally semi-concave while ρ 0 is a finite Borel measure. The structure of the velocity and the density is exhibited. For initial data correlated in a certain sense, a class of classical solutions (S, ρ) is given. Negative time is also considered, and illustrating examples are given.

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Section: Outline and Principal Resultsmentioning

confidence: 68%

“…Sections 3 through 6 are devoted to the viscosity-measure solution (S, ρ) of the initial-value problem (6,7).…”

Section: Outline and Principal Resultsmentioning

confidence: 99%

“…This means that we investigate viscosity solutions [8] of (6) and measure solutions [14] of (7). This combination of generalized solutions was recently extensively studied in [1].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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A One-Dimensional Inviscid and Compressible Fluid in a Harmonic Potential Well

Choquard

Strömberg

2007

Acta Appl Math

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“…2 , −t (coupling const./temperature), with Weiss's theory of magnetism is particularly striking if, for the τ = 0 case, we fold in the first quadrant the curve: magnetization versus magnetic field [17,Eqn. 23].…”

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confidence: 87%

The inviscid, compressible and rotational, 2D isotropic Burgers and pressureless Euler–Coriolis fluids: Solvable models with illustrations

Choquard

Vuffray²

2014

Physica D: Nonlinear Phenomena

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The coupling between dilatation and vorticity, two coexisting and fundamental processes in fluid dynamics [1, pp. 3, 6] is investigated here, in the simplest cases of inviscid 2D isotropic Burgers and pressureless Euler-Coriolis fluids respectively modeled by single vortices confined in compressible, local, inertial and global, rotating, environments. The field equations are established, inductively, starting from the equations of the characteristics solved with an initial Helmholtz decomposition of the velocity fields namely a vorticity free and a divergence free part [1, Sects. 2.3.2, 2.3.3] and, deductively, by means of a canonical Hamiltonian Clebsch like formalism [2], [3], implying two pairs of conjugate variables. Two vector valued fields are constants of the motion: the velocity field in the Burgers case and the momentum field per unit mass in the Euler-Coriolis one. Taking advantage of this property, a class of solutions for the mass densities of the fluids is given by the Jacobian of their sum with respect to the actual coordinates. Implementation of the isotropy hypothesis entails a radial dependence of the velocity potentials and of the stream functions associated to the compressible and to the rotational part of the fluids and results in the cancellation of the dilatation-rotational cross terms in the Jacobian. A simple expression is obtained for all the radially symmetric Jacobians occurring in the theory. Representative examples of regular and singular solutions are shown and the competition between dilatation and vorticity is illustrated. Inspired by thermodynamical, mean field theoretical analogies, a genuine variational formula is proposed which yields unique measure solutions for the radially symmetric fluid densities investigated. We stress that this variational formula, unlike the Hopf-Lax formula, enables us to treat systems which are both compressible and rotational. Moreover in the one-dimensional case, we show for an interesting application that both variational formulas are equivalent.

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Non-Convex Multipartite Ferromagnets

Genovese

Tantari

2016

J Stat Phys

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Abstract. We investigate a multipartite ferromagnetic model without self-interactions between spins of the same party, so that the Hamiltonian is not a definite quadratic form of the magnetisations. We find the free energy and study the phase transition for all zero external fields. Moreover in the bipartite case we analyse the fluctuations of the rescaled magnetisations, below and at the critical point, and we study the phase transitions with non-zero magnetic fields.

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On the “Mean Field” Interpretation of Burgers' Equation

Cited by 20 publications

References 3 publications

A One-Dimensional Inviscid and Compressible Fluid in a Harmonic Potential Well

A One-Dimensional Inviscid and Compressible Fluid in a Harmonic Potential Well

The inviscid, compressible and rotational, 2D isotropic Burgers and pressureless Euler–Coriolis fluids: Solvable models with illustrations

Non-Convex Multipartite Ferromagnets

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