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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. MSC: 35Q35, 76N10key words: steady compressible Navier-Stokes system, shallow water equation, low Mach number limit, density dependent viscosities, large data, existence via Schauder type fixed point theorem.

The Navier-Stokes-Fourier system is a well established model for describing the motion of viscous compressible heat-conducting fluids. We study the existence of time-periodic weak solutions and improve the result from [3] in the following sense: we extend the class of pressure functions (i.e. consider lower exponent γ) and include also the effect of radiation on the boundary.

We consider a model describing the steady flow of compressible heat-conducting chemically reacting multicomponent mixture. We show the existence of strong solutions under the additional assumption that the mixture is sufficiently dense. We work in the L p -setting combining the methods for the weak solutions with the method of decomposition. The result is a generalization of previous results of the authors, where the case of single-constituted fluid was studied.

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