C. Lienstromberg and J. J. L. Velázquez, “Long-time asymptotics and regularity estimates for weak solutions to a doubly degenerate thin-film equation in the Taylor-Couette setting.” arXiv, to appear in Pure and Applied Analysis, 2023. doi:
10.48550/ARXIV.2203.00075.
Abstract
We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the sum of its first and third spatial derivative. The modelling equation appears as a thin-film limit for the interface separating two immiscible viscous fluid films confined between two cylinders rotating at small relative angular velocity. More precisely, the fluid occupying the layer next to the outer cylinder is considered to be Newtonian, i.e. it has constant viscosity, while we assume that the layer next to the inner cylinder is filled by a shear-thinning power-law fluid.
Using energy methods, Fourier analysis and suitable regularity estimates for higher-order parabolic equations, we prove global existence of positive weak solutions in the case of low initial energy. Moreover, these global solutions are polynomially stable, in the sense that interfaces which are initially close to a circle, converge at rate 1/t1/β for some β>0 to a circle, as time tends to infinity.
In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.BibTeX
P. Gladbach, J. Jansen, and C. Lienstromberg, “Non-Newtonian thin-film equations: global existence of solutions, gradient-flow structure and guaranteed lift-off,” 2023, doi:
10.48550/ARXIV.2301.10300.
Abstract
We study the gradient-flow structure of a non-Newtonian thin film equation with power-law rheology. The equation is quasilinear, of fourth order and doubly-degenerate parabolic. By adding a singular potential to the natural Dirichlet energy, we introduce a modified version of the thin-film equation. Then, we set up a minimising-movement scheme that converges to global positive weak solutions to the modified problem. These solutions satisfy an energy-dissipation equality and follow a gradient flow. In the limit of a vanishing singularity of the potential, we obtain global non-negative weak solutions to the power-law thin-film equation
∂tu+∂x(m(u)|∂3xu−G′′(u)∂xu|α−1(∂3xu−G′′(u)∂xu))=0
with potential G in the shear-thinning (α>1), Newtonian (α=1) and shear-thickening case (0<α<1). The latter satisfy an energy-dissipation inequality. Finally, we derive dissipation bounds in the case G≡0 which imply that solutions emerging from initial values with low energy lift up uniformly in finite time.BibTeX
C. Lienstromberg, S. Schiffer, and R. Schubert, “A data-driven approach to viscous fluid mechanics: The stationary case,”
Arch. Rational Mech. Anal., vol. 247, no. 30, Art. no. 30, 2023, doi:
10.1007/s00205-023-01849-w.
Abstract
We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid’s viscosity in the mathematical model, we suggest directly using experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics (Kirchdoerfer and Ortiz in Comput Methods Appl Mech Eng 304:81–101, 2016; Conti et al. in Arch Ration Mech Anal 229:79–123, 2018). We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and -quasiconvexity, we show a -convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply not only to inertialess fluids but also to fluids with inertia. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.BibTeX
C. Lienstromberg, S. Schiffer, and R. Schubert, “A variational approach to the non-newtonian Navier-Stokes equations,” 2023. doi:
doi:10.48550/ARXIV.2312.03546.
Abstract
We present a variational approach for the construction of Leray-Hopf solutions to the non-newtonian Navier-Stokes system. Inspired by the work OSS18 on the corresponding Newtonian problem, we minimise certain stabilised Weighted Inertia-Dissipation-Energy (WIDE) functionals and pass to the limit of a vanishing parameter in order to recover a Leray-Hopf solution of the non-newtonian Navier-Stokes equations. It turns out that the results differ depending on the rheology of the fluid. The investigation of the non-newtonian Navier-Stokes system via this variational approach is motivated by the fact that it is particularly well suited to gain insights into weak, respectively strong convergence properties for different flow-behaviour exponents and thus into possibly turbulent behaviour of the fluid flow. ...BibTeX
O. Assenmacher, G. Bruell, and C. Lienstromberg, “Non-Newtonian two-phase thin-film problem: local existence, uniqueness, and stability,”
Comm. Partial Differential Equations, vol. 47, no. 1, Art. no. 1, 2022, doi:
10.1080/03605302.2021.1957929.
Abstract
Summary: "We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the twop-hase Navier-Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.''BibTeX
J. Jansen, C. Lienstromberg, and K. Nik, “Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order.” arXiv, 2022. doi:
10.48550/ARXIV.2204.08231.
Abstract
We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the H1(Ω)-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate 1/t1/β for some β>0. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in H1(Ω).BibTeX
C. Lienstromberg, T. Pernas-Castano, and J. J. L. Velázquez, “Analysis of a two-fluid Taylor-Couette flow with one non-Newtonian fluid,”
J. Nonlinear Sci., vol. 32, no. 2, Art. no. 2, 2022, doi:
10.1007/s00332-021-09750-0.
Abstract
Summary: "We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier-Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.''BibTeX
J. Escher, P. Knopf, C. Lienstromberg, and B.-V. Matioc, “Stratified periodic water waves with singular density gradients,”
Ann. Mat. Pura Appl. (4), vol. 199, no. 5, Art. no. 5, 2020, doi:
10.1007/s10231-020-00950-1.
Abstract
The authors consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. Three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients are established. The availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis are used. The paper is organized as follows. In Section 2, the three formulations of the problem are introduced and their equivalence is established. In Section 3, the authors first introduce the notion of a weak solution to Dubreil-Jacotin's formulation and establish, by means of a shooting method, the existence of at least one laminar flow solution to this latter formulation. In Section 4, the equations are reformulated as an abstract bifurcation problem by using methods from nonlinear functional analysis.BibTeX
C. Lienstromberg and S. Müller, “Local strong solutions to a quasilinear degenerate fourth-order thin-film equation,”
NoDEA Nonlinear Differential Equations Appl., vol. 27, no. 2, Art. no. 2, 2020, doi:
10.1007/s00030-020-0619-x.
Abstract
In this paper, the authors study a model describing the evolution of the height of a non-Newtonian incompressible thin liquid film that spreads on a solid surface by the Ellis constitutive law. Existence of strong positive solutions in Sobolev-Slobodetskiĭ spaces is shown by using Amann's abstract existence theory. In addition, the authors prove uniqueness of the solutions by the energy approach.BibTeX
J. Escher and C. Lienstromberg, “Travelling waves in dilatant non-Newtonian thin films,”
J. Differential Equations, vol. 264, no. 3, Art. no. 3, 2018, doi:
10.1016/j.jde.2017.10.015.
Abstract
In this article, the authors prove the existence of a traveling wave solution for a gravity-driven thin film of a viscous incompressible dilatant fluid coated with an insoluble surfactant. The governing system of second-order partial differential equations for the film's height h and the surfactant's concentration γ is derived by means of lubrication theory applied to the non-Newtonian Navier-Stokes equations. (Note that there exists a well-known ansatz for modeling non-Newtonian fluids see, for example, D. Klimov and A. Petrov, Arch. Appl. Mech. 70 (2000), no. 1-3, 3–16, doi:10.1007/s004199900027.)
The general ansatz and the estimation of the energy of the wave solution presented here will be of interest for people working in the specific field. The present article is an excellent development in this area of research.BibTeX
J. Escher, P. Gosselet, and C. Lienstromberg, “A note on model reduction for microelectromechanical systems,”
Nonlinearity, vol. 30, no. 2, Art. no. 2, 2017, doi:
10.1088/1361-6544/aa4ff9.
Abstract
"Numerical evidence is provided that there are non-constant permittivity profiles which force solutions to a two-dimensional coupled moving boundary problem modelling microelectromechanical systems to be positive, while the corresponding small-aspect ratio model produces solutions which are always non-positive.''BibTeX
C. Lienstromberg, “Well-posedness of a quasilinear evolution problem modelling MEMS with general permittivity,”
J. Evol. Equ., vol. 17, no. 4, Art. no. 4, 2017, doi:
10.1007/s00028-016-0375-x.
Abstract
"Subject of consideration is the analytical investigation of a coupled system of partial differential equations arising from the modelling of electrostatically actuated microelectromechanical systems with general permittivity profile. A quasilinear parabolic evolution problem for the displacement u of an elastic membrane is coupled with an elliptic free boundary value problem that determines the electrostatic potential ψ in the region between the elastic membrane and a rigid ground plate. The system is shown to be well-posed locally in time for all arbitrarily large values λ of the applied voltage, whereas small values of the applied voltage, which do not exceed a certain critical value λ∗, do even allow globally in time existing solutions. In addition, conditions are specified which force solutions emerging from a non-positive initial deflection to stay non-positive as long as they exist.''BibTeX
J. Escher and C. Lienstromberg, “A survey on second-order free boundary value problems modelling MEMS with general permittivity profile,”
Discrete Contin. Dyn. Syst. Ser. S, vol. 10, no. 4, Art. no. 4, 2017, doi:
10.3934/dcdss.2017038.
Abstract
"In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi- or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles.
"The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account—the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.''BibTeX
J. Escher and C. Lienstromberg, “Finite-time singularities of solutions to microelectromechanical systems with general permittivity,”
Ann. Mat. Pura Appl. (4), vol. 195, no. 6, Art. no. 6, 2016, doi:
10.1007/s10231-016-0549-8.
BibTeX
J. Escher and C. Lienstromberg, “A qualitative analysis of solutions to microelectromechanical systems with curvature and nonlinear permittivity profile,”
Comm. Partial Differential Equations, vol. 41, no. 1, Art. no. 1, 2016, doi:
10.1080/03605302.2015.1105259.
BibTeX
C. Lienstromberg, “On qualitative properties of solutions to microelectromechanical systems with general permittivity,”
Monatsh. Math., vol. 179, no. 4, Art. no. 4, 2016, doi:
10.1007/s00605-015-0744-5.
Abstract
"Qualitative properties of solutions to the evolution problem modelling microelectromechanical systems with general permittivity profile are investigated. The system couples a parabolic evolution problem for the displacement of a membrane with an elliptic free boundary value problem for the electric potential in the region between the membrane and a rigid ground plate. We briefly allude to results concerning local and global well-posedness and the small-apect ratio limit. However, the focus is here on proving non-positivity of the membrane displacement for the full moving boundary problem under certain boundary conditions on the potential, as well as the existence of finite-time singularities assuming to have a non-positive solution.''BibTeX
C. Lienstromberg, “A free boundary value problem modelling microelectromechanical systems with general permittivity,”
Nonlinear Anal. Real World Appl., vol. 25, pp. 190--218, 2015, doi:
10.1016/j.nonrwa.2015.03.008.
Abstract
Än investigation of the free boundary value problem arising from the modelling of electrostatically actuated microelectromechanical systems with general permittivity is presented. Consisting of a parabolic evolution problem for the displacement of a membrane as well as of an elliptic moving boundary problem for the electric potential between the membrane and a rigid ground plate, the system is shown to be well-posed locally in time for all values λ of the applied voltage. It is in addition verified that the solution exists even globally in time, provided that the applied voltage does not exceed a certain critical value λ∗. Furthermore, we establish the convergence of the solution of the free boundary problem towards the solution of the small-aspect ratio model, as the aspect ratio tends to zero.''BibTeX
