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

Abstract

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.

Steady solutions to a model of compressible chemically reacting fluid with high density

“…In contrast to the single-constituted fluid case studied in Axmann et al 4 or Axmann et al, 5 we are not able to deduce here the comparison principle for the temperature. Therefore, inspired by previous works on compressible mixtures and in order to guarantee the nonnegativity of temperature as well as the specific concentrations, we introduce two approximation parameters , > 0.…”

“…[6][7][8]19 Note also that our approach is also based on the method of decomposition, developed and successfully used in nineties to study strong solutions of the compressible Navier-Stokes equations by Novotný and Padula, see previous study. 20 This fact will be hidden here as we can directly use results from Axmann et al 4 and Axmann et al, 5 which are based on this technique, and we do not have to prove them here again.…”

“…for arbitrary 1 < p < ∞. We may replace the condition on geometric properties of Ω by assuming that f > 0 and f ∼ M. The proof follows similar lines, only in order to estimate the velocity, we need to combine the estimates from the entropy and the total energy balance and the proof becomes slightly more complicated, see also Axmann et al 4 and Axmann et al 5 Remark 1. Integrating Equation (4), by virtue of the boundary conditions ( 16) and (18), one can observe that any stationary solution to our system has to satisfy sort of compatibility conditions…”

“…In order to improve the regularity we use the method of decomposition in the same way as for the case of single-component fluid in Axmann et al 4 and Axmann et al 5 Here, we use the fact that the momentum equation is influenced by the chemical reactions only through the temperature involved in the pressure. Otherwise, all the other terms in the momentum equation are identical to the single-component case, and thus can be treated exactly in the same way.…”

Steady solutions to a model of compressible chemically reacting fluid with high density

“…We may replace the condition on geometric properties of Ω by assuming that f > 0 and f ∼ M . The proof follows similar lines, only in order to estimate the velocity, we need to combine the estimates from the entropy and the total energy balance and the proof becomes slightly more complicated, see also [2] and [3].…”