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Discrete and Continuous Dynamical Systems - S

November 2021 , Volume 14 , Issue 11

Issue on partial differential equations in fluids and solids

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Helmut Abels, Karoline Disser, Hans-Christoph Kaiser, Alexander Mielke and Marita Thomas
2021, 14(11): i-iii doi: 10.3934/dcdss.2021101 +[Abstract](3369) +[HTML](346) +[PDF](108.93KB)
Discrete approximation of dynamic phase-field fracture in visco-elastic materials
Marita Thomas and Sven Tornquist
2021, 14(11): 3865-3924 doi: 10.3934/dcdss.2021067 +[Abstract](1132) +[HTML](353) +[PDF](737.55KB)

This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively \begin{document}$ 1 $\end{document}-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys
Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček and Petr Sedlák
2021, 14(11): 3925-3952 doi: 10.3934/dcdss.2020459 +[Abstract](1575) +[HTML](743) +[PDF](1490.2KB)

A phenomenological model for polycrystalline NiTi shape-memory alloys with a refined dissipation function is here enhanced by a thermomechanical coupling and rigorously analyzed as far as existence of weak solutions and numerical stability and convergence of the numerical approximation performed by a staggered time discretization. Moreover, the model is verified on one-dimensional computational simulations compared with real laboratory experiments on a NiTi wire.

Dimension reduction of thermistor models for large-area organic light-emitting diodes
Annegret Glitzky, Matthias Liero and Grigor Nika
2021, 14(11): 3953-3971 doi: 10.3934/dcdss.2020460 +[Abstract](1456) +[HTML](458) +[PDF](418.7KB)

An effective system of partial differential equations describing the heat and current flow through a thin organic light-emitting diode (OLED) mounted on a glass substrate is rigorously derived from a recently introduced fully three-dimensional \begin{document}$ p(x) $\end{document}-Laplace thermistor model. The OLED consists of several thin layers that scale differently with respect to the multiscale parameter \begin{document}$ \varepsilon>0 $\end{document}, which is the ratio between the total thickness and the lateral extent of the OLED. Starting point of the derivation is a rescaled formulation of the current-flow equation in the OLED for the driving potential and the heat equation in OLED and glass substrate with Joule heat term concentrated in the OLED. Assuming physically motivated scalings in the electrical flux functions, uniform a priori bounds are derived for the solutions of the three-dimensional system which facilitates the extraction of converging subsequences with limits that are identified as solutions of a dimension reduced system. In the latter, the effective current-flow equation is given by two semilinear equations in the two-dimensional cross-sections of the electrodes and algebraic equations for the continuity of the electrical fluxes through the organic layers. The effective heat equation is formulated only in the glass substrate with Joule heat term on the part of the boundary where the OLED is mounted.

On a linearized Mullins-Sekerka/Stokes system for two-phase flows
Helmut Abels and Andreas Marquardt
2021, 14(11): 3973-3987 doi: 10.3934/dcdss.2020467 +[Abstract](1242) +[HTML](428) +[PDF](423.85KB)

We study a linearized Mullins-Sekerka/Stokes system in a bounded domain with various boundary conditions. This system plays an important role to prove the convergence of a Stokes/Cahn-Hilliard system to its sharp interface limit, which is a Stokes/Mullins-Sekerka system, and to prove solvability of the latter system locally in time. We prove solvability of the linearized system in suitable \begin{document}$ L^2 $\end{document}-Sobolev spaces with the aid of a maximal regularity result for non-autonomous abstract linear evolution equations.

Cahn–Hilliard–Brinkman systems for tumour growth
Matthias Ebenbeck, Harald Garcke and Robert Nürnberg
2021, 14(11): 3989-4033 doi: 10.3934/dcdss.2021034 +[Abstract](1558) +[HTML](393) +[PDF](1468.72KB)

A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn–Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman–Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.

A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion
Pierre-Etienne Druet
2021, 14(11): 4035-4067 doi: 10.3934/dcdss.2020458 +[Abstract](1714) +[HTML](455) +[PDF](479.48KB)

After the pioneering work by Giovangigli on mathematics of multicomponent flows, several attempts were made to introduce global weak solutions for the PDEs describing the dynamics of fluid mixtures. While the incompressible case with constant density was enlighted well enough due to results by Chen and Jüngel (isothermal case), or Marion and Temam, some open questions remain for the weak solution theory of gas mixtures with their corresponding equations of mixed parabolic–hyperbolic type. For instance, Mucha, Pokorny and Zatorska showed the possibility to stabilise the hyperbolic component by means of the Bresch-Desjardins technique and a regularisation of pressure preventing vacuum. The result by Dreyer, Druet, Gajewski and Guhlke avoids ex machina stabilisations, but the mathematical assumption that the Onsager matrix is uniformly positive on certain subspaces leads, in the dilute limit, to infinite diffusion velocities which are not compatible with the Maxwell-Stefan form of diffusion fluxes. In this paper, we prove the existence of global weak solutions for isothermal and ideal compressible mixtures with natural diffusion. The main new tool is an asymptotic condition imposed at low pressure on the binary Maxwell-Stefan diffusivities, which compensates possibly extreme behaviour of weak solutions in the rarefied regime.

From quasi-incompressible to semi-compressible fluids
Tomáš Roubíček
2021, 14(11): 4069-4092 doi: 10.3934/dcdss.2020414 +[Abstract](1706) +[HTML](544) +[PDF](477.76KB)

A new concept of semi-compressible fluids is introduced for slightly compressible visco-elastic fluids (typically rather liquids than gasses) where mass density variations are negligible in some sense, while being directly controlled by pressure which is very small in comparison with the elastic bulk modulus. The physically consistent fully Eulerian models with specific dispersion of pressure-wave speed are devised. This contrasts to the so-called quasi-incompressible fluids which are described not physically consistently and, in fact, only approximate ideally incompressible ones in the limit. After surveying and modifying models for the quasi-incompressible fluids, we eventually devise some fully convective models complying with energy conservation and capturing phenomena as pressure-wave propagation with wave-length (and possibly also pressure) dependent velocity.

Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions
Hannes Eberlein and Michael Růžička
2021, 14(11): 4093-4140 doi: 10.3934/dcdss.2020419 +[Abstract](1429) +[HTML](567) +[PDF](635.79KB)

We consider a viscous, incompressible Newtonian fluid flowing through a thin elastic (non-cylindrical) structure. The motion of the structure is described by the equations of a linearised Koiter shell, whose motion is restricted to transverse displacements. The fluid and the structure are coupled by the continuity of velocities and an equilibrium of surface forces on the interface between fluid and structure. On a fixed in- and outflow region we prescribe natural boundary conditions. We show that weak solutions exist as long as the shell does not self-intersect.

Equilibrium of immersed hyperelastic solids
Manuel Friedrich, Martin Kružík and Ulisse Stefanelli
2021, 14(11): 4141-4157 doi: 10.3934/dcdss.2021003 +[Abstract](1349) +[HTML](387) +[PDF](385.95KB)

We discuss different equilibrium problems for hyperelastic solids immersed in a fluid at rest. In particular, solids are subjected to gravity and hydrostatic pressure on their immersed boundaries. By means of a variational approach, we discuss free-floating bodies, anchored solids, and floating vessels. Conditions for the existence of local and global energy minimizers are presented.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6

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