Advances in Differential Equations

Some smoothness and uniqueness results for a shallow-water problem

Franois Joseph Chatelon and Pierre Orenga

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In a previous work, we have shown the existence of weak solutions for a shallow-water problem (or compressible two-dimensional Navier-Stokes problem) in a depth-mean velocity formulation. Some results have been proved by Kazhikov in the case of domain equal to $\mathbb{R}^N$ and linearized momentum equation, which allows him to look for a velocity of the form $u=\nabla p$. We present some smoothness and uniqueness results whatever a smooth domain and with the complete momentum equation and boundary conditions on $u \cdot n$ and $\mathcal{u}$.

Article information

Adv. Differential Equations, Volume 3, Number 1 (1998), 155-176.

First available in Project Euclid: 19 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35M10: Equations of mixed type 35Q35: PDEs in connection with fluid mechanics 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30] 76M30: Variational methods


Chatelon, Franois Joseph; Orenga, Pierre. Some smoothness and uniqueness results for a shallow-water problem. Adv. Differential Equations 3 (1998), no. 1, 155--176.

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