Communications in Mathematical Sciences

Volume 15 (2017)

Number 5

Steady solutions to viscous shallow water equations. The case of heavy water.

Pages: 1385 – 1402



Šimon Axmann (Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic)

Piotr Bogusław Mucha (Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland)

Milan Pokorný (Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic)


In this note, we show the existence of regular solutions to the stationary version of the Navier–Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods for weak solutions to the compressible Navier–Stokes system.


steady compressible Navier–Stokes system, shallow water equation, low Mach number limit; density-dependent viscosities, large data, existence via Schauder-type fixed point theorem

2010 Mathematics Subject Classification

35Q35, 76N10

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The work on this paper was partially conducted during the first author’s internship at the Warsaw Center of Mathematics and Computer Science.

The first and the third author were supported by Czech Science Foundation (grant no. 16-03230S). The second author (PBM) has been partly supported by National Science Centre grant 2014/14/M/ST1/00108 (Harmonia).

Received 13 July 2016

Published 26 June 2017