Global solutions of two-dimensional Navier-Stokes and euler equations

Matania Ben-Artzi

doi:10.1007/BF00387712

Global solutions of two-dimensional Navier-Stokes and euler equations

Matania Ben-Artzi^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

88 Scopus citations

Abstract

Long-time solutions to the Navier-Stokes (NS) and Euler (E) equations of incompressible flow in the whole plane are constructed, under the assumption that the initial vorticity is in L¹(ℝ²) for (NS) and in L¹(ℝ²)∩ L^r(ℝ²) for some r>2 for (E). It is shown that the solution to (NS) is unique, smooth and depends continuously on the initial data, and that the (velocity) solution to (E) is Hölder continuous in the space and time coordinates. It is shown that as the viscosity vanishes, there is a subsequence of solutions to (NS) converging to a solution of (E).

Original language	English
Pages (from-to)	329-358
Number of pages	30
Journal	Archive for Rational Mechanics and Analysis
Volume	128
Issue number	4
DOIs	https://doi.org/10.1007/BF00387712
State	Published - Dec 1994

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abstract = "Long-time solutions to the Navier-Stokes (NS) and Euler (E) equations of incompressible flow in the whole plane are constructed, under the assumption that the initial vorticity is in L1(ℝ2) for (NS) and in L1(ℝ2)∩ Lr(ℝ2) for some r>2 for (E). It is shown that the solution to (NS) is unique, smooth and depends continuously on the initial data, and that the (velocity) solution to (E) is H{\"o}lder continuous in the space and time coordinates. It is shown that as the viscosity vanishes, there is a subsequence of solutions to (NS) converging to a solution of (E).",

author = "Matania Ben-Artzi",

year = "1994",

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doi = "10.1007/BF00387712",

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N2 - Long-time solutions to the Navier-Stokes (NS) and Euler (E) equations of incompressible flow in the whole plane are constructed, under the assumption that the initial vorticity is in L1(ℝ2) for (NS) and in L1(ℝ2)∩ Lr(ℝ2) for some r>2 for (E). It is shown that the solution to (NS) is unique, smooth and depends continuously on the initial data, and that the (velocity) solution to (E) is Hölder continuous in the space and time coordinates. It is shown that as the viscosity vanishes, there is a subsequence of solutions to (NS) converging to a solution of (E).

AB - Long-time solutions to the Navier-Stokes (NS) and Euler (E) equations of incompressible flow in the whole plane are constructed, under the assumption that the initial vorticity is in L1(ℝ2) for (NS) and in L1(ℝ2)∩ Lr(ℝ2) for some r>2 for (E). It is shown that the solution to (NS) is unique, smooth and depends continuously on the initial data, and that the (velocity) solution to (E) is Hölder continuous in the space and time coordinates. It is shown that as the viscosity vanishes, there is a subsequence of solutions to (NS) converging to a solution of (E).

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Global solutions of two-dimensional Navier-Stokes and euler equations

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