Uniqueness of a solution to the Cauchy problem for the Hopf equation in the two-dimensional case
type: Item
instance of: scholarly article
title: Uniqueness of a solution to the Cauchy problem for the Hopf equation in the two-dimensional case
zbMATH Open document ID: 1229.35179
author: V. I. Gishlarkaev
published in: Journal of Mathematical Sciences (New York)
publication date: 2011-06-30T00:00:00Z
review text: It is a very interesting paper, concerning the statistical approach of 2D turbulent flow, described by the Hopf equation. The Cauchy problem for this equation was studied by \textit{C. Foias} [Rend. Sem. Mat. Univ. Padova 48(1972), 219--348 (1973; Zbl 0283.76017)], where the existence and uniqueness were obtained, assuming that an integral related with the flow energy (containing an exponential of the solution) is finite. In the present paper, the generalized Fubini theorem for a transition measure is used to obtain the uniqueness, when the mean energy of the initial condition is finite. No additional conditions are necessary, therefore an improvement of Foias's result is obtained. The Galerkin approximation is used to solve the abstract analog of the Navier-Stokes equations in a separable Hilbert space.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 5915269
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Hopf equation
zbMATH Keywords: turbulence
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: uniqueness
reviewed by: Gelu Paşa
MaRDI profile type: Publication
full work available at URL: s10958-010-0039-2
OpenAlex ID: W2067588397
cites work: Q3785226
cites work: SOME MATHEMATICAL PROBLEMS OF STATISTICAL HYROMECHANICS
cites work: Q5817196
cites work: Identification problems for a steady-state model of mass transfer
cites work: Q4163866
cites work: Q4937653
cites work: Q4177788
DOI: 10.1007/S10958-010-0039-2
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description: scientific article; zbMATH DE number 5915269