arXiv subjects

Miroslav Bulicek

Publications and source records attributed to Miroslav Bulicek.

2 recordsLinked to original sources

arXiv2024

On the existence of solutions to generalized Navier--Stokes--Fourier system with dissipative heating

We consider a flow of non-Newtonian incompressible heat conducting fluids with dissipative heating. Such system can be obtained by scaling the classical Navier--Stokes--Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck--Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck--Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.

math.AP↗

arXiv2018

Well posedness of nonlinear parabolic systems beyond duality

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system \[ \partial_tu-\mathrm{div} ( ν(|\nabla u|) \nabla u )= -\mathrm{div} f \] with a given {strictly} positive bounded function $ν$, {such that $\lim_{k\to \infty} ν(k)=ν_\infty$} and $f \in L^q$ with $q\in (1,\infty)$. The {existence, uniqueness and regularity} results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty)$.

math.AP↗