Page 400 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 400
MULTICOMPONENT DIFFUSION IN POROUS MEDIA (MS-42)
Existence analysis of a stationary compressible fluid model for
heat-conducting and chemically reacting mixtures
Milan Pokorny, milan.pokorny@mff.cuni.cz
Charles University, Czech Republic
Coauthors: Miroslav Bulicek, Ansgar Juengel, Nicola Zamponi
The existence of large-data weak solutions to a steady compressible Navier–Stokes–Fourier
system for chemically reacting fluid mixtures is proved. General free energies are considered
satisfying some structural assumptions, with a pressure containing a γ-power law. The model
is thermodynamically consistent and contains the Maxwell–Stefan cross-diffusion equations in
the Fick–Onsager form as a special case. Compared to previous works, a very general model
class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids,
and different molar masses. A priori estimates are derived from the entropy balance and the
total energy balance. The compactness for the total mass density follows from an estimate for
the pressure in Lp with p > 1, the effective viscous flux identity, and uniform bounds related
to Feireisl’s oscillations defect measure. These bounds rely heavily on the convexity of the free
energy and the strong convergence of the relative chemical potentials.
Uniqueness for a cross-diffusion system issuing from seawater intrusion
problems
Carole Rosier, rosier@univ-littoral.fr
Université du Littoral, France
Coauthor: Catherine Choquet
This work is devoted to the mathematical analysis of the Cauchy problem for cross-diffusion
systems without any assumption about its entropic structure. A global existence result of non-
negative solutions is obtained by applying a classical Schauder fixed point theorem. The proof
is upgraded for enhancing the regularity of the solution, namely its gradient belongs to the space
Lr((0, T )×Ω) for some r > 2. To this aim, the Schauder’s strategy is coupled with an extension
of Meyers regularity result for linear parabolic equations. We show how this approach allows to
prove the well-posedness of the problem using only assumptions prescribing and admissibility
range for the ratios between the diffusion and cross-diffusion coefficients. Finally, the question
of the maximal principle is also addressed, especially when source terms are incorporated in the
equation in order to ensure the confinement of the solution.
Variational methods for fluid-structure interaction: Bulk elastic solids
interacting with the Navier Stokes equation
Sebastian Schwarzacher, schwarz@karlin.mff.cuni.cz
Charles University, Prague, Czech Republic
Coauthors: Barbora Benesova, Malte Kampschulte
We introduce a two time-scale scheme which allows to extend the method of minimizing move-
ments to hyperbolic problems. This method is used to show the existence of weak solutions to
a fluid-structure interaction problem between a nonlinear, visco-elastic, n-dimensional bulk
398
Existence analysis of a stationary compressible fluid model for
heat-conducting and chemically reacting mixtures
Milan Pokorny, milan.pokorny@mff.cuni.cz
Charles University, Czech Republic
Coauthors: Miroslav Bulicek, Ansgar Juengel, Nicola Zamponi
The existence of large-data weak solutions to a steady compressible Navier–Stokes–Fourier
system for chemically reacting fluid mixtures is proved. General free energies are considered
satisfying some structural assumptions, with a pressure containing a γ-power law. The model
is thermodynamically consistent and contains the Maxwell–Stefan cross-diffusion equations in
the Fick–Onsager form as a special case. Compared to previous works, a very general model
class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids,
and different molar masses. A priori estimates are derived from the entropy balance and the
total energy balance. The compactness for the total mass density follows from an estimate for
the pressure in Lp with p > 1, the effective viscous flux identity, and uniform bounds related
to Feireisl’s oscillations defect measure. These bounds rely heavily on the convexity of the free
energy and the strong convergence of the relative chemical potentials.
Uniqueness for a cross-diffusion system issuing from seawater intrusion
problems
Carole Rosier, rosier@univ-littoral.fr
Université du Littoral, France
Coauthor: Catherine Choquet
This work is devoted to the mathematical analysis of the Cauchy problem for cross-diffusion
systems without any assumption about its entropic structure. A global existence result of non-
negative solutions is obtained by applying a classical Schauder fixed point theorem. The proof
is upgraded for enhancing the regularity of the solution, namely its gradient belongs to the space
Lr((0, T )×Ω) for some r > 2. To this aim, the Schauder’s strategy is coupled with an extension
of Meyers regularity result for linear parabolic equations. We show how this approach allows to
prove the well-posedness of the problem using only assumptions prescribing and admissibility
range for the ratios between the diffusion and cross-diffusion coefficients. Finally, the question
of the maximal principle is also addressed, especially when source terms are incorporated in the
equation in order to ensure the confinement of the solution.
Variational methods for fluid-structure interaction: Bulk elastic solids
interacting with the Navier Stokes equation
Sebastian Schwarzacher, schwarz@karlin.mff.cuni.cz
Charles University, Prague, Czech Republic
Coauthors: Barbora Benesova, Malte Kampschulte
We introduce a two time-scale scheme which allows to extend the method of minimizing move-
ments to hyperbolic problems. This method is used to show the existence of weak solutions to
a fluid-structure interaction problem between a nonlinear, visco-elastic, n-dimensional bulk
398
