Occurrence and non-appearance of shocks in fractal Burgers equations

Nathael Alibaud, Jerome Droniou, Julien Vovelle

Research output: Contribution to journalArticleResearchpeer-review

51 Citations (Scopus)

Abstract

We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently large initial conditions, by giving a result which states that, for smooth small initial data, the solution remains at least Lipschitz continuous.
Original languageEnglish
Pages (from-to)479 - 499
Number of pages21
JournalJournal of Hyperbolic Differential Equations
Volume4
Issue number3
DOIs
Publication statusPublished - 2007
Externally publishedYes

Cite this

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Occurrence and non-appearance of shocks in fractal Burgers equations. / Alibaud, Nathael; Droniou, Jerome; Vovelle, Julien.

In: Journal of Hyperbolic Differential Equations, Vol. 4, No. 3, 2007, p. 479 - 499.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Alibaud, Nathael

AU - Droniou, Jerome

AU - Vovelle, Julien

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AB - We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently large initial conditions, by giving a result which states that, for smooth small initial data, the solution remains at least Lipschitz continuous.

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DO - 10.1142/S0219891607001227

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