Special Session 2: New frontiers in the compressible mathematical fluid mechanics and thermodynamics

Existence results for thermoviscoelastic fluid models

Miroslav Bul\\'{\\i}\\v{c}ek Charles University Czech Rep Co-Author(s):

Abstract: We present theoretical results on two classes of thermodynamically consistent models for heat-conducting viscoelastic fluids: (i) rate-type models of Giesekus type, and (ii) kinetic models based on the Fokker–Planck equation for dilute polymeric solutions. For both frameworks, we establish the existence of global weak solutions under natural assumptions on initial data and demonstrate their stability properties. The analysis rigorously captures key aspects of viscoelastic and thermal behavior, including energy storage and entropy production, providing a solid foundation for the study of complex materials with coupled viscous, elastic, and thermal effects.

Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model

Nilasis Chaudhuri University of Warsaw Poland Co-Author(s):    Christian Rohde, Florian Wendt

Abstract: In this talk, we explore a relaxation model for a compressible fluid with capillarity effects in an isothermal setting. The model involves two parameters that control the strength of the relaxation and are expected to drive the system toward the classical compressible Navier--Stokes--Korteweg equations in an appropriate limiting regime. We introduce a notion of finite-energy weak solutions for the associated initial--boundary value problem in three spatial dimensions. Within this framework, we establish a weak--strong uniqueness result. In addition, we provide a first rigorous justification of the relaxation model by proving convergence to the target system in the relaxation limit at the level of finite-energy weak solutions.

Reverse in time blowup for compressible NSE

Mimi Dai University of Illinois at Chicago USA Co-Author(s):

Abstract: We will discuss a construction for the compressible Navier-Stokes equations that produces solutions blowing up from the right in time. This is a different blowup phenomenon from other known blowup scenarios for the compressible Navier-Stokes equations.

Mach, Rossby and the Rayleigh-B\`enard problem

Francesco Fanelli Basque Center for Applied Mathematics Spain Co-Author(s):

Abstract: In this talk, we revisit some recent results about the dynamics of heat conducting fluids in the incompressible limit regime, under non-homogeneous boundary conditions for the temperature (a relevant situation in the context of the celebrated Rayleigh-B\`enard problem). We show how the classical Oberbeck-Boussinesq system, usually identified as the target system, must be modified in this case, or even dismissed in presence of additional fast rotation effects. This talk is based on joint works with Eduard Feireisl (Czech Academy of Sciences).

On a system of equations arising in meteorology: Well-posedness and data assimilation

Piotr Gwiazda Institute of Mathematics Polish Academy of Sciences Poland Co-Author(s):

Abstract: Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a singular limit of the three-dimensional compressible Navier-Stokes-Fourier system with rotation driven by temperature gradient. The limit system preserves the essential physical mechanisms of the original model, while exhibiting a reduced, effectively two-and-a-half-dimensional structure. This simplified framework allows for a rigorous analytical study of the data assimilation process while maintaining a direct physical connection to the full compressible model. We establish well posedness of global-in-time solutions and a compact trajectory attractor, followed by the stability and convergence results for the nudging scheme applied to the limiting system. Finally, we demonstrate how these results can be combined with a relative entropy argument to extend the assimilation framework to the full three-dimensional compressible setting, thereby establishing a rigorous connection between the reduced and physically complete models.

Dafermos entropy criterion and dissipative weak solutions of the compressible Euler equations

Maria Lukacova Institute of Mathematics, University Mainz Germany Co-Author(s):    Eduard Feireisl, Changsheng Yu

Abstract: The dissipative solutions of the Euler equations can be obtained as a limit of suitable structure-preserving, consistent, and stable numerical schemes. If the strong solution to the above equations exists, the dissipative solutions coincide with the strong solution on its life span. Otherwise, we apply the concept of K-convergence and prove the strong convergence of the empirical means of numerical solutions to a dissipative weak solution. The latter is the expected value of the dissipative measure-valued solutions and satisfies a weak formulation of the Euler equations, modulo the Reynolds turbulent stress. In this talk, we will discuss the relationship between numerical approximations of oscillatory solutions and selection criteria, such as the Dafermos criterion, which maximises entropy production, or the optimisation of the energy defect. A series of numerical simulations will illustrate theoretical results.

Unconditional stability of equilibria in thermally driven compressible fluids

Yong Lyu Nanjing University Peoples Rep of China Co-Author(s):    Eduard Feireis, Yongzhong Sun

Abstract: We show that small perturbations of the spatially homogeneous equilibrium of a thermally driven compressible viscous fluid are globally stable. Specifically, any weak solution of the evolutionary Navier--Stokes--Fourier system driven by thermal convection converges to an equilibrium as time goes to infinity. The main difficulty to overcome is the fact the problem does not admit any obvious Lyapunov function. The result applies, in particular, to the Rayleigh-Benard convection problem.

Stability and convergence in compressible magnetohydrodynamics

Hana Mizerova Comenius University in Bratislava Slovak Rep Co-Author(s):

Abstract: This talk presents recent results on the analysis of the full compressible magnetohydrodynamic system with non-conservative boundary conditions, providing a rigorous bridge between analytical theory and computational practice. Dissipative measure-valued (DMV) solutions are introduced together with a DMV-strong uniqueness principle, ensuring that DMV solutions coincide with strong solutions as long as the latter exist. This stability result provides a robust foundation for numerical analysis. In particular, a structure-preserving finite volume scheme is considered, and the DMV-strong uniqueness principle is employed to prove convergence of numerical solutions to the physically relevant strong solution.

Various approaches to the issue of existence for systems of compressible flows

Piotr B Mucha University of Warsaw Poland Co-Author(s):

Abstract: In this talk, I will discuss several currently available techniques for proving the existence of weak solutions to systems describing compressible flows. The presentation will focus on methods based on the construction of solutions, with particular emphasis on their main ideas, limitations, and the open problems that remain in the field. Special attention will be given to approaches relying on the renormalization of the continuity equation. Methods based on convex integration will not be considered. The aim of the talk is to provide a comparative overview of existing analytical frameworks and to highlight directions for future research in the mathematical theory of compressible fluid models.

Steady compressible Navier--Stokes--Fourier system with temperature dependent viscosities

Milan Pokorny Charles University Czech Rep Co-Author(s):    Ondrej Kreml, Tomasz Piasecki, Emil Skrisovsky

Abstract: We consider the model describing the steady flow of a compressible heat conducting fluid in a bounded three-dimensional domain \[ \begin{array}{c} \displaystyle \mbox{div}\, (\varrho \mathbf{u}) = 0, \ \displaystyle \mbox{div}\, (\varrho \mathbf{u} \otimes \mathbf{u}) - \mbox{div}\, \mathbf{S} + \nabla p = \varrho \mathbf{f}, \ \mbox{div}\, (\varrho E \mathbf{u}) = \varrho \mathbf{f} \cdot \mathbf{u} - \mbox{div}\, (p \mathbf{u}) + \mbox{div}\, (\mathbf{S} \mathbf{u}) -\mbox{div}\, \mathbf{q} \end{array} \] with \(\varrho\) the density, \(\mathbf{u}\) the velocity field, \(\mathbf{S}\) the stress tensor (here we assume the fluid to be Newtonian with temperature dependent viscosity), \(p\) the pressure, \(\mathbf{f}\) the given volume force, \(\mathbf{q}\) the heat flux and the total energy \(E= \frac 12 |\mathbf{u}|^2 + e\) with \(e\) the internal energy. We assume the pressure law of the form \(p(\varrho, \vartheta) \sim \varrho^{\gamma} + \varrho \vartheta\) with \(\gamma>1\) and the viscosities \(\mu(\vartheta), \xi (\vartheta) \sim (1+ \vartheta)^\alpha\), \(\alpha \in [0,1]\). We show the existence of a weak or variational entropy solution for the above model with internal energy fulfilling the Gibbs relation and the heat flux fulfilling the Fourier law \(\mathbf{q} \sim (1+\vartheta)^m \nabla \vartheta\) with \(\vartheta\) the temperature, \(m = m(\gamma, \alpha)>0\). We first review the results for the case $\alpha=1$ which has been known before. Then we show extensions for $\alpha \frac 32$ both for Dirichlet and Navier boundary conditions for the velocity and Robin and Dirichlet boundary conditions for the temperature. Finally we briefly touch the situation when $\gamma \leq \frac 32$, here only for Robin boundary conditions for the temperature. The solutions are constructed for arbitrarily large sufficiently integrable data.

The hydrostatic Lagrangian approach to the compressible primitive equations

Arnab Roy Basque Center for Applied Mathematics (BCAM) Spain Co-Author(s):    Matthias Hieber, Yoshiki Iida and Tarek Z\{o}chling

Abstract: In this talk, we explain the hydrostatic Lagrangian approach to the compressible primitive equations. A fundamental aspect in the analysis is the investigation of the compressible hydrostatic Lam\`{e} and Stokes operators. Local strong well-posedness for large data and global strong well-posedness for small data are established under various assumptions on the pressure law, both in the presence and absence of gravity.

A finite element method for the incompressible Euler system

Bangwei She Capital Normal University Peoples Rep of China Co-Author(s):    M\`{a}ria Luk\`{a}\v{c}ov\`{a}-Medvid`ov\`{a}

Abstract: We study the stability and convergence of a $RT_k/P_k$ finite element method for the incompressible Euler system. Moreover, we extend the error analysis result of Guzm\`{a}n et al. [H(div) conforming and DG methods for incompressible Euler equations, IMA J. Numer. Anal. 37(4)1733-1771, 2017] to the case of k=0.

Unconditional stability of radially symmetric steady sates of compressible viscous fluids with inflow/outflow boundary conditions

Agnieszka Swierczewska-Gwiazda University of Warsaw Poland Co-Author(s):

Abstract: We show that certain radially symmetric steady states of compressible viscous fluids in domains with inflow/outflow boundary conditions are unconditionally stable. This means that any not necessarily radially symmetric solution of the associated evolutionary problem converges to a single radially symmetric steady state.

Incompressible limits at large Mach number for a reduced compressible MHD system

Aneta Wr\`oblewska-Kami\`nska Institute of Mathematics, Polish Academy of Sciences Poland Co-Author(s):

Abstract: In this talk we will present a singular limit problem for a reduced model for compressible non-resistive MHD. This system can also be related to a certain class of two-fluid models. By a suitable rescaling of the magnetic pressure in terms of some parameter $\varepsilon > 0$, by letting $\varepsilon \to 0$ we perform the incompressible limit while keeping the Mach number of order $O(1)$. Our study is conducted in the framework of global in time finite energy weak solutions and for ill-prepared initial data. We also consider a similar problem in presence of a strong Coriolis term. The key ingredient of the proof, based on a compensated compactness argument, is the use of the transport equation (well-known in the context of two-fluid models) underlying the dynamics. Thanks to it, and differently from previous studies about the incompressible limit, we are able to identify the asymptotic of the terms of order $O(\varepsilon)$ and to characterise their dynamics; such an information is in fact crucial to obtain a closed system in the limit. This is recent joint result with Francesco Fanelli and Young-Sam Kwon.

temperature-driven turbulence in compressible fluid flows

Yuhuan Yuan Nanjing University of Aeronautics and Astronautics Peoples Rep of China Co-Author(s):

Abstract: We study the long-time behaviour of the temperature-driven compressible flows. We show that numerical solutions of a structure-preserving finite volume method generate a discrete attractor that consists of entire discrete trajectories. Further, we prove the convergence of discrete attractors to their continuous counterparts. Theoretical results are illustrated by extensive numerical simulations of the well-known Rayleigh-Benard problem. The numerical results also indicate the validity of the ergodic hypothesis and imply that a non-zero Reynolds stress persist for long time. Finally, we also observe that any invariant measure is of Gaussian type in sharp contrast with the conjecture proposed by [Glimm et al., SN Applied Sciences, 2:paper no.2160, 2020].

Relative entropy method and applications for the compressible two-fluid flows

Ewelina Zatorska University of Warwick England Co-Author(s):    Yang Li, Maria Lukacova-Medvidova, Milan Pokorny

Abstract: I will present a model of compressible two viscous fluids flow with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotn\`{y} and Pokorn\`{y} [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech., 2022]. In this talk I will focus on the weak--strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of one of phases obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.