Special Session 96: Recent Trends in Navier-Stokes Equations, Euler Equations, and Related Problems

$\boldsymbol L^p$-theory for the operator $\text{\bf curl\,}(\alpha\,\text{\bf curl\,}\cdot)$ and applications

Cherif Amrouche Universite de Pau et des Pays de l`Adour France Co-Author(s):    Lisa Santo

Abstract: In this work, we develop an $L^p$-theory for the existence of weak and strong solutions to the elliptic problem $$-\text{div\,}(a\nabla u)=f,$$ in a bounded open domain of $\mathbb R^N$, under Dirichlet or Neumann boundary conditions, and under suitable assumptions on the data and on the regularity of the domain boundary. We also develop an $\boldsymbol L^p$-theory for the existence of weak and strong solutions to the vector-valued problem defined in a bounded open multiply connectde domain of $\mathbb R^3$, $$\text{\bf curl\,}(\alpha \text{\bf curl\,}\boldsymbol h)=\boldsymbol f,$$ considering both perfectly permeable and perfectly conducting boundary conditions, and different assumptions on $\boldsymbol f$ and on the regularity of $\partial\Omega$. As an application, we study a coupled electromagnetic induction heating problem, obtaining improved regularity results for the solutions compared to those available in the existing literature.

Inviscid incompressible limit of the compressible Navier-Stokes system with a moving rigid body

Kuntal Bhandari Institute of Mathematics of the Czech Academy of Sciences Czech Rep Co-Author(s):    \v{S}\`{a}rka Ne\v{c}asov\`{a}, Arnab Roy

Abstract: We present the inviscid incompressible limit of a system of compressible Navier-Stokes equations interacting with a moving rigid body. Both cases of no-slip and Navier-slip condition on the fluid-rigid body interface will be considered.

On the stability of steady states for a generalized Navier-Stokes-Fourier system

Petr Kaplicky Charles University Czech Rep Co-Author(s):    Karol Hajduk, Aneta Wroblewska

Abstract: We consider the flow of a generalized non-Newtonian incompressible heat-conducting fluid in a bounded domain $\Omega \subset \mathbb{R}^3$, subject to Dirichlet boundary conditions for both velocity and temperature. While we assume homogeneous Dirichlet boundary conditions for the velocity, which prohibit the exchange of mass with the surroundings, we allow nonhomogeneous Dirichlet boundary conditions for the temperature, which enforces heat exchange with the surroundings. The fluid obeys a power-law constitutive relation for the Cauchy stress, and the thermal conductivity is assumed to be a nonlinear function of the temperature, which may degenerate or become unbounded. No external forces are considered. We study the long-time behaviour of the system. We show that the steady state of the system is nonlinearly stable within a suitably defined class of solutions. We also prove the existence of such solutions. This is follow-up research to the work done together with A. Abbatiello and M. Bul\`\i\v cek. It is joint work with Aneta Wroblevska and Karol Hajduk.

Time-periodic solutions to incompressible flows in moving domains

Ondrej Kreml Institute of Mathematics, Czech Academy of Sciences Czech Rep Co-Author(s):

Abstract: We study the flow of an incompressible viscous fluid in a domain which changes shape periodically in time. We show that this problem admits a time-periodic weak solution. We are able to remove all smallness assumptions which where imposed in the previous results.

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams

Vaclav Macha Institute of Mathematics, Czech Academy of Sciences Czech Rep Co-Author(s):

Abstract: We study a nonlinear fluid-structure interaction problem between a viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions. Under the effect of periodic forces acting on the beam and the fluid, at least one time-periodic weak solution is constructed which has a bounded energy and a fixed prescribed mass.

Dynamic interaction between a rigid-body and an incompressible viscous fluid: some new results for the IBVP

Paolo Maremonti Universiit\`{a} degli Studi della Campania ``L. Vanvitelli`` Italy Co-Author(s):

Abstract: We investigate some analytic questions concerning the solutions to the equations of a model related to the dynamical interaction between a rigid-body and a viscous incompressible fluid. The model is a generalization of Oseen problem related to the ordinary Navier-Stokes equations. Actually, on its surface $\partial B$, the rigid body B is forced by the Newtonian stress tensor, that is the dynamical answer of the incompressible fluid. We are able to discuss some questions of uniqueness of regular solutions, asymptotic properties of the solutions and a possible structure theorem related to a suitable weak solution. All the questions are studied by considering the IBVP in a frame which is attached to the rigid body. This frame from one side simplifies the approach to the existence of solutions, from another side it makes the inconvenient to introduce the velocity of the rigid motion inside the equations as a coefficient. In its entirety the problem of the existence of regular solutions was solved in 2002, the case of global solutions for small data in 2023, instead the other questions, that are the uniqueness, asymptotic behavior of the kinetic energy and partial regularity of a weak solution, they were open problems until 2025.

Thermal effects in fluid structure interactions

Sourav Mitra IIT Indore India Co-Author(s):

Abstract: In this talk we consider two different heat conducting fluids each modeled by the incompressible Navier-Stokes-Fourier system separated by a non-linear elastic Koiter shell. The motion of the shell changes the domain of definition of the two separated fluids. For this setting we show the existence of a weak solution. We follow a variational approach for fluid-structure interactions. To include temperature a novel two step minimization scheme is used to produce an approximation. The weak solutions are energetically closed and include a strictly positive temperature.

Regular Solutions in Fluid Mechanics: Choosing Between Besov and Sobolev Frameworks

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

Abstract: The analysis of regular solutions to fluid mechanics equations typically begins with a concession: we assume smallness conditions on the initial data. This is not a choice of convenience, but a mathematical necessity that allows us to treat the system as a perturbation of a linear problem. Once this regime is established, the central technical challenge becomes the selection of an appropriate functional framework. This talk examines the practical trade-offs between Besov and Sobolev spaces when constructing these solutions. While Besov spaces offer refined tools for frequency localization and critical scaling -- often providing better results in the whole space $R^n$ --they frequently become a liability when dealing with physical boundaries. Conversely, Sobolev spaces, while perhaps less sharp in a scaling sense, provide a more robust environment for handling boundary conditions and integration by parts. Through specific examples in both unbounded and bounded domains, I will discuss the advantages and disadvantages of each approach. The goal is to illustrate that the best space is rarely universal, but rather depends on whether one prioritizes frequency-side precision or physical-space boundary compatibility.

The primitive equations and conjecture of Onsager

Sarka Necasova Czech Academy of Sciences Czech Rep Co-Author(s):    M.A. Rodriqez-Bellido, T. Tang, , E. Wiedeman, L. Zhu

Abstract: In the talk, we will focus on the problem of the conservation of energy for the weak solutions to the compressible primitive equations system with degenerate viscosity. This part of the talk is based on collaboration with M.A. Bellido and T. Tang. Moreover, we will consider the Osanger conjecture for the incompressible primitive system in a bounded domain. This part of the talk is based on the collaborations with E. Wiedemann, T. Tang, and L. Zhu.

Long time density-dependent Navier-Stokes equations with large flux

Joanna Renc{\\l}awowicz Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology Poland Co-Author(s):    Wojciech Zaj\c{a}czkowski

Abstract: The nonhomogeneous (density-dependent) Navier-Stokes equations are considered in a cylindrical domain in $R^3$, parallel to the $x_3$-axis with large inflow and outflow on the top and the bottom. Moreover, on the lateral part of the cylinder the slip boundary conditions are assumed. The long-time existence of regular solutions is proved under assumptions that inflow and outflow are close to homogeneous and norms of derivatives with respect to $x_3$ of the external force and initial velocity are sufficiently small. The key point is to verify that the $x_3$-coordinate of velocity remains positive.

Divergence-Free Dual Spaces and $L^p$ Analysis for the Stokes and Navier--Stokes Equations with Mixed Navier-Type Boundary Conditions

Nour Seloula University of Caen Normandie France Co-Author(s):    Imane Boussetouan and Cherif Amrouche

Abstract: In this talk, we study the stationary Stokes and Navier--Stokes equations in an $L^p$ framework under mixed Navier-type boundary conditions, combining Dirichlet conditions on part of the boundary with prescribed normal trace and tangential vorticity on the complementary part. A central aspect of this work is the precise characterization of the dual spaces associated with divergence-free function spaces adapted to these boundary conditions. This duality framework allows for a rigorous variational formulation beyond the Hilbert setting. As an application, we establish the existence of weak and strong solutions in the $L^p$ theory for both the Stokes and Navier--Stokes systems under these mixed boundary conditions.

On the Navier-Stokes equations with energy-stable outflow boundary conditions

Ana Silvestre Instituto Superior T\`ecnico, Universidade de Lisboa Portugal Co-Author(s):

Abstract: We are interested in flow of a viscous incompressible fluid in distorted pipes of finite length, modeled through the Navier-Stokes equations with mixed boundary conditions. At the outlets, directional do-nothing boundary conditions are imposed on the fluid motion. We investigate existence, uniqueness and regularity of solutions in the $2d$ and $3d$ cases. This is joint work with Alessio Falocchi (Politecnico di Milano, Italy) and Gianmarco Sperone (Pontificia Universidad Cat\`{o}lica de Chile, Santiago).

Long-time behavior of solutions to fluid dynamic shape optimization problems via phase-field method

John Sebastian H Simon University of Koblenz Germany Co-Author(s):    Michael Hinze and Christian Kahle

Abstract: We investigate the long time behavior of solutions to a shape and topology optimization problem with respect to the time-dependent Navier--Stokes equations. The sought topology is represented by a stationary phase-field that represents a smooth indicator function. The fluid equations are approximated by a porous media approach and are time-dependent. In the latter aspect, the considered problem formulation extends earlier work. We prove that if the time horizon tends to infinity, minima of the time-dependent problem converge towards minima of the corresponding stationary problem. To do so, a convergence rate with respect to the time horizon, of the values of the objective functional, is analytically derived. This allowed us to prove that the solution to the time-dependent problem converges to a phase-field, as the time horizon goes to infinity, which is proven to be a minimizer for the stationary problem. We validate our results by numerical investigation.

Some remarks on Leray`s structure theorem

Werner Varnhorn Institute of Mathematics, Kassel University Germany Co-Author(s):

Abstract: We consider a general weak solution of the nonstationary nonlinear Navier-Stokes equations and extend some results concerning Leray`s structure theorem by omitting the strong energy inequality.

Coupled Vlasov and non-Newtonian dynamics

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

Abstract: We study a coupled kinetic-non-Newtonian fluid system on the periodic domain $\mathcal{T}^3$, where particles evolve by a Vlasov equation and interact with an incompressible power-law fluid through a drag force. We prove the global existence of weak solutions for all $p > \frac{8}{5}$, where $p > 1$ denotes the power-law exponent of the fluid`s stress-strain relation. Under an additional uniform boundedness assumption on the particle density, we also establish large-time decay of a modulated energy functional measuring deviation from velocity alignment. The decay rate is algebraic when $p > 2$ and exponential when $\frac{6}{5} \leq p \leq 2$, reflecting the role of fluid dissipation in the large-time dynamics. This is recent joint work with Young-Pil Choi and Jinwook Jung.