Special Session 195: Calculus of Variations and Hyperbolic PDEs in Solid Mechanics

Low Regularity Potentials in Heterogeneous Cahn--Hilliard Functionals

Jakob Deutsch TU Wien Austria Co-Author(s):    Riccardo Cristoferi, Luca Pignatelli

Abstract: We investigate the Cahn--Hilliard functional, a prototypical model for liquid--liquid phase separation, in a highly irregular setting. In particular, we consider potentials of low regularity that vanish on space-dependent wells. Under very mild assumptions, we prove a robust compactness result. By slightly strengthening the regularity of the wells and the growth conditions of the potential near them, we then fully characterize the asymptotic behavior of the associated family of functionals via a $\Gamma$-convergence analysis.

Variational approaches to inertia, contact and limits

Malte Kampschulte Charles University Prague, Faculty of Mathematics and Physics Czech Rep Co-Author(s):

Abstract: Variational approaches to inertia, contact and limits For static and quasi-static problems, (iterated) minimization has long been one of the most important tools to prove existence of solutions. The main advantage of these variational approaches is that they able to deal with complicated nonlinearities and nonconvexities in a rather natural fashion, directly relying on the description of a problem in terms of its physical energy. In contrast, for dynamic problems, i.e. those involving inertia, such variational approaches so far have been much less useful in practical existence proofs. The aim of this talk is to present our recent and not so recent attempts at bridging this gap, using a time-delayed approach which uses energetical descriptions and minimization as both a modelling approach, as well as a way of showing existence of solutions. This will be illustrated in the example of viscoelastodynamics, where we can deal with situations such as contact that defy any approach based on linearization. Furthermore we will see how the same ideas can be used to study limit systems of parameter-dependent families of such problems in a similarly general fashion. This is based on joint works with, among others, B.Bene\v{s}ov\`a, A.\v{C}e\v{s}\`ik, G.Gravina, M.Kru\v{z}\`ik and S.Schwarzacher.

Geometric rigidity in variable domains and applications in dimension reduction

Leonard Kreutz Technical University of Munich Germany Co-Author(s):    Manuel Friedrich, Konstantinos Zemas

Abstract: In this talk we present a quantitative geometric rigidity estimate in dimensions $d=2,3$ generalising a celebrated result by Friesecke, James and M\uller to the setting of variable domains. Loosely speaking, we show that for each function $y\in H^1(U;\mathbb{R}^3)$ and for each connected component of an open bounded set $U \subset \mathbb{R}^d$, the $L^2$-distance of $\nabla y$ from a single rotation can be controlled up to a constant by its $L^2$-distance from the group $SO(d)$, with the constant not depending on the precise shape of $U$, but only on an integral curvature functional related to $\partial U$. We further show that for linear strains the estimate can be refined, leading to a uniform control independent of the set $U$. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation (GSBD) for sequences of displacements related to deformations with uniformly bounded elastic energy. We show how this estimate can be applied in the context of dimension reduction by calculating the $\Gamma$-limits for thin elastic solids containing voids in different energy scaling regimes in terms of their thickness. This is seminar based on joint work on with Manuel Friedrich (Johannes Kepler Universit\at Linz) and Konstantinos Zemas (Universit\at Bonn).

Frame-indifferent approximation in Nonlinear Thermoviscoelasticity

Lennart Machill Rheinische Friedrich-Wilhelms-Universit\\"{a}t Bonn Germany Co-Author(s):

Abstract: We consider a Kelvin-Voigt model for thermoviscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame indifference. We prove the existence of weak solutions relying on an estimate due to Ciarlet and Mardare (2015), which generalizes the celebrated rigidity estimate of Friesecke, James, and M\uller (2002). The estimate allows us to employ a staggered time-discretization scheme that preserves frame indifference, in which the deformation and the temperature are updated alternately. The talk is based on joint works with Manuel Friedrich, Martin Kru\v{z}\`ik, Rufat Badal, and Martin Hor\`{a}k.

Derivation of membrane models in nonlocal hyperelasticity

Anastasia Molchanova TU Wien Austria Co-Author(s):    Dominik Engl, Hidde Sch\onberger

Abstract: Motivated by the analysis of thin structures, we study the variational dimension reduction of hyperelastic energies involving nonlocal gradients to an effective membrane model. When rescaling the thin domain, initially isotropic interaction ranges naturally become anisotropic, which leads us to develop a theory of anisotropic nonlocal gradients with direction-dependent interaction ranges. In contrast to existing nonlocal derivatives with finite horizon, which are typically defined via interaction kernels supported on balls of positive radius, our formulation employs ellipsoidal interaction regions whose principal radii may vanish independently. This yields a unified framework interpolating between fully nonlocal, partially nonlocal, and purely local models. Using these tools, we establish a $\Gamma$-convergence analysis for the associated nonlocal thin-film energies. The limit functional retains the structural form of the classical membrane energy, and the classical local model is recovered exactly when all interaction radii vanish. This talk is based on joint work with Dominik Engl and Hidde Sch\onberger.

Curvature-driven pattern formation in biomembranes: A gradient flow approach

Anastasija Pesic WIAS Berlin Germany Co-Author(s):    Patrik Knopf, Dennis Trautwein

Abstract: In this talk, we will present a phase-field model for curvature-driven pattern formation in biomembranes. The model is obtained as a gradient flow of an energy that approximates the two-phase Canham--Helfrich functional, which leads to a Cahn--Hilliard-type equation with cross diffusion for the lipid concentration, coupled to a fourth-order equation for the membrane height. We establish existence and uniqueness of weak solutions for this system, both for regular and for singular potentials. In addition, we present numerical simulations that demonstrate how different parameter regimes give rise to striped, dotted, or snake-like patterns on the membrane. The talk is based on a joint work with Patrik Knopf and Dennis Trautwein.

An existence theorem for time-dependent obstacle problems in viscoelasticity

Paolo Piersanti The Chinese University of Hong Kong, Shenzhen Peoples Rep of China Co-Author(s):

Abstract: We establish the existence of solutions for a model describing the evolution of a linearly viscoelastic body which is constrained to remain confined in a prescribed half-space. The confinement condition under consideration is of Signorini type, and is given over the boundary of the linearly viscoelastic body under consideration. We show that one such variational problem admits solutions and we coin a novel concept of solution which, differently from the available literature, is valid even in the case where the viscoelastic body starts its motion in contact with the obstacle. Additionally, under additional assumptions on the constituting material, we show that when the applied body force is lifted the deformed linearly viscoelastic body returns to its rest position at an exponential rate of decay.

Time-Periodic Solutions for Hyperbolic-Parabolic Systems

Sebastian Schwarzacher Uppsala University/Charles University Sweden Co-Author(s):    Stanislav Mosny, Boris Muha, Justin T. Webster

Abstract: Time-periodic weak solutions for a coupled hyperbolic-parabolic system are obtained. A linear heat and wave equation are considered on two respective d-dimensional spatial domains that share a common (d-1)-dimensional interface. The system is only partially damped, leading to an indeterminate case for existing theory. We construct periodic solutions by obtaining novel a priori estimates for the coupled system, reconstructing the total energy via the interacting interface. As a byproduct, geometric constraints manifest on the wave domain which are reminiscent of classical boundary control conditions for wave stabilizability. We note a ``loss of regularity between the forcing and solution which is greater than that associated with the heat-wave Cauchy problem. However, we consider a broader class of spatial domains and mitigate this regularity loss by trading time and space differentiations, a feature unique to the periodic setting. This seems to be the first constructive result addressing existence and uniqueness of periodic solutions in the heat-wave context, where no dissipation is present in the wave interior.

Global Weak Solutions for Korteweg-Type Fluid Models

Stefano Spirito University of L Aquila Italy Co-Author(s):

Abstract: We present recent results on the global-in-time existence of finite-energy weak solutions for some compressible fluid models with capillarity, allowing for density-dependent and possibly degenerate viscosity coefficients. The analysis combines energy methods with entropy structures and compactness techniques, yielding a robust existence theory under minimal regularity assumptions.

Regular solutions and long-time dynamics in nonlinearly coupled thermoelasticity

Srdan Trifunovic Faculty of Sciences, University of Novi Sad Yugoslavia Co-Author(s):    Piotr Michal Bies,Tomasz Cieslak, Mario Fuest, Johannes Lankeit, Boris Muha

Abstract: In this talk, I will present a recent result on a nonlinearnly coupled thermoelasticity system. The first part of the result concerns existence and uniqueness of regular solutions, while the second one concerns long-time dynamics. The result is based on a novel functional involving the Fisher information associated with temperature, and its boundedness plays a key role in the result.

Regular solutions and long-time dynamics in nonlinearly coupled thermoelasticity

Srdan Trifunovic Faculty of Sciences, University of Novi Sad Yugoslavia Co-Author(s):    Piotr Micha{\l} Bies,Tomasz Cie\`slak, Mario Fuest, Johannes Lankeit, Boris Muha

Abstract: In this talk, I will present a recent result on a nonlinearnly coupled thermoelasticity system. The first part of the result concerns existence and uniqueness of regular solutions, while the second one concerns long-time dynamics. The result is based on a novel functional involving the Fisher information associated with temperature, and its boundedness plays a key role in the result.

Stability effect in elasticity

Dehua Wang University of Pittsburgh USA Co-Author(s):

Abstract: Elasticity is important in continuum mechanics with a wide range of applications and is challenging in analysis. In this talk we shall discuss some special elastic effects in elastic fluids, including the stabilizing effect of elasticity on the vortex sheets in compressible elastic flows and on the vanishing viscosity process of compressible viscoelastic flows.

Energy barriers for boundary nucleation in solid-solid phase transitions

Konstantinos Zemas University of Bonn Germany Co-Author(s):    Antonio Tribuzio

Abstract: We study energy-scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary nucleation of a single-phase martensite inside a parental phase of austenite, our main focus is how the relationship between the rank-1 direction and the orientation of the half-space influences the energy scaling with respect to the fixed volume of the inclusion. Up to prefactors depending on this relative orientation, the scaling laws coincide with the corresponding ones for bulk nucleation, for all rank-1 directions, but the ones normal to the confining hyperplane. In the latter, the scaling is as in a three-well problem in full space, resulting in a lower energy barrier. This is joint work with A. Tribuzio (Uni-Bonn).

Rate dependent dislocation dynamics and gradient flows of currents

Anton\`in \v Ce\v s\`ik University of Warwick England Co-Author(s):    Thomas Hudson, Malte Kampschulte, Filip Rindler

Abstract: Dislocations are line defects in crystalline materials whose dissipative motion drives plastic deformation. Their evolution involves complex topological changes and the formation of singular geometries. In this talk, we propose a variational framework for rate-dependent dislocation dynamics with superlinear dissipation. Dislocations are modeled as integral currents, which naturally capture the underlying geometric singularities. The evolution is formulated as a generalized gradient flow of currents. A central difficulty is that the dissipation is not lower semicontinuous in the natural weak${}^*$ topology, due to the rate dependence. To overcome this, we introduce an additional Young-measure structure for the velocity field, which captures the spatial micro-oscillations arising in the limit. The evolution is constructed via the Weighted-Energy-Dissipation (WED) variational approach, leading to solutions satisfying an energy--dissipation inequality.