arXiv subjects

Gieri Simonett

Publications and source records attributed to Gieri Simonett.

At least 19 recordsLinked to original sources

arXiv2026

On nematic electrolytes

We study a system of nonlinear partial differential equations modeling the electrokinetics of a nematic electrolyte material consisting of various ion species suspended in a nematic liquid crystal within a bounded domain in two or three dimensions. The system couples a Nernst-Planck model for ion concentrations with the Poisson equation for the electrostatic potential, a Navier-Stokes equation for the fluid solvent, and the Ericksen-Leslie equations with general Leslie stress for nematic liquid crystals. We consider the case of isotropic elasticity for the liquid crystal and impose a unit-length constraint on the director field. The no-flux condition for the electrochemical potential leads to a nonlinear (and nonlocal) boundary condition for the ion concentrations. Using the theory of maximal regularity, we prove existence and uniqueness of strong solutions, provide criteria for global existence, and characterize the set of equilibria.

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arXiv2024

Coriolis-driven fluid motion on spherical surfaces

We consider the motion of an incompressible viscous fluid on a sphere, incorporating the effects of the Coriolis force. We demonstrate that global solutions exist for any divergence-free initial condition with finite kinetic energy. Furthermore, we show that each solution converges at an exponential rate to a state that is aligned with the rotation of the sphere.

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arXiv2023

The Navier-Stokes equations on manifolds with boundary

We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold $\sM$ with boundary. The motion on $\sM$ is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on $\partial\sM$. We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on $\sM$ that satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case $\sM$ is two-dimensional we show that solutions with divergence free initial condition in $L_2(\sM; T\sM)$ exist globally and converge to an equilibrium exponentially fast.

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arXiv2023

On a thermodynamically consistent model for magnetoviscoelastic fluids in 3D

We introduce a system of equations that models a non-isothermal magnetoviscoelastic fluid. We show that the model is thermodynamically consistent, and that the critical points of the entropy functional with prescribed energy correspond exactly with the equilibria of the system. The system is investigated in the framework of quasilinear parabolic systems and shown to be locally well-posed in an $L_p$-setting. Furthermore, we prove that constant equilibria are normally stable. In particular, we show that solutions that start close to a constant equilibrium exist globally and converge exponentially fast to a (possibly different) constant equilibrium. Finally, we establish that the negative entropy serves as a strict Lyapunov functional and we then show that every solution that is eventually bounded in the topology of the natural state space exists globally and converges to the set of equilibria.

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arXiv2022

Well-posedness for magnetoviscoelastic fluids in 3D

We show that the system of equations describing a magnetoviscoelastic fluid in three dimensions can be cast as a quasilinear parabolic system. Using the theory of maximal $L_p$-regularity, we establish existence and uniqueness of local strong solutions and we show that each solution is smooth (in fact analytic) in space and time. Moreover, we give a complete characterization of the set of equilibria and show that solutions that start out close to a constant equilibrium exist globally and converge to a (possibly different) constant equilibrium. Finally, we show that every solution that is eventually bounded in the topology of the state space exists globally and converges to the set of equilibria.

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arXiv2021

$H^\infty$-calculus for the surface Stokes operator and applications

We consider a smooth, compact and embedded hypersurface $Σ$ without boundary and show that the corresponding (shifted) surface Stokes operator $ω+A_{S,Σ}$ admits a bounded $H^\infty$-calculus with angle smaller than $π/2$, provided $ω>0$. As an application, we consider critical spaces for the Navier-Stokes equations on the surface $Σ$. In case $Σ$ is two-dimensional, we show that any solution with a divergence-free initial value in $L_2(Σ,\mathsf{T}Σ)$ exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field.

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arXiv2020

On the Navier-Stokes equations on surfaces

We consider the motion of an incompressible viscous fluid that completely covers a smooth, compact and embedded hypersurface $Σ$ without boundary and flows along $Σ$. Local-in-time well-posedness is established in the framework of $L_p$-$L_q$-maximal regularity. We characterize the set of equilibria as the set of all Killing vector fields on $Σ$ and we show that each equilibrium on $Σ$ is stable. Moreover, it is shown that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to infinity.

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arXiv2019

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth-order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well-posedness of both flows for initial surfaces that are $C^{1+α}$-regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long-term existence for initial surfaces which are $C^{1+α}$-close to a sphere, and we prove that these solutions become spherical as time goes to infinity.

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arXiv2018

On the motion of a fluid-filled rigid body with Navier Boundary conditions

We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution à la Leray-Hopf to the equations of motion converges to an equilibrium at an exponential rate in the $L_q$-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology.

math.AP↗

arXiv2018

On quasilinear parabolic equations and continuous maximal regularity

We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.

math.AP↗

arXiv2018

The Rayleigh-Taylor instability for the Verigin problem with and without phase transition

Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh-Taylor instability.

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arXiv2018

The Verigin problem with and without phase transition

Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria are identified and their thermodynamical stability is investigated by means of a variational approach. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. It is further shown that a non-degenerate equilibrium is dynamically stable in the natural state manifold if and only if it is thermodynamically stable. Finally, it is shown that a solution which does not develop singularities exists globally and converges to an equilibrium in the state manifold.

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arXiv2018

A maximal regularity approach to the study of motion of a rigid body with a fluid-filled cavity

We consider the inertial motion of a rigid body with an interior cavity that is completely filled with a viscous incompressible fluid. The equilibria of the system are characterized and their stability properties are analyzed. It is shown that equilibria associated with the largest moment of inertia are normally stable, while all other equilibria are normally hyperbolic. We show that every Leray-Hopf weak solution converges to an equilibrium at an exponential rate. In addition, we determine the critical spaces for the governing evolution equation, and we demonstrate how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity of solutions and their convergence to equilibria.

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arXiv2018

On the microscopic bidomain problem with FitzHugh-Nagumo ionic transport

The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is studied in the $L_p\!-\!L_q$-framework. Reformulating the problem as a semilinear evolution equation on the interface, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension $d\leq 3$, by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria parallel those of the classical FitzHugh-Nagumo system in ODE's. These properties of the bidomain equations are obtained combining recent results on Dirichlet-to-Neumann operators, on critical spaces for parabolic evolution equations, and qualitative theory of evolution equations.

math.AP↗

arXiv2017

Critical spaces for quasilinear parabolic evolution equations and applications

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier-Stokes problem, convection-diffusion equations,the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.

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arXiv2016

Stability of equilibrium shapes in some free boundary problems involving fluids

In this paper the motion of two-phase, incompressible, viscous fluids with surface tension is investigated. Three cases are considered: (1) the case of heat-conducting fluids, (2) the case of isothermal fluids, and (3) the case of Stokes flows. In all three situations, the equilibrium states in the absence of outer forces are characterized and their stability properties are analyzed. It is shown that the equilibrium states correspond to the critical points of a natural physical or geometric functional (entropy, available energy, surface area) constrained by the pertinent conserved quantities (total energy, phase volumes). Moreover, it is shown that solutions which do not develop singularities exist globally and converge to an equilibrium state.

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arXiv2016

On the Muskat flow

Of concern is the motion of two fluids separated by a free interface in a porous medium, where the velocities are given by Darcy's law. We consider the case with and without phase transition. It is shown that the resulting models can be understood as purely geometric evolution laws, where the motion of the separating interface depends in a non-local way on the mean curvature. It turns out that the models are volume preserving and surface area reducing, the latter property giving rise to a Lyapunov function. We show well-posedness of the models, characterize all equilibria, and study the dynamic stability of the equilibria. Lastly, we show that solutions which do not develop singularities exist globally and converge exponentially fast to an equilibrium.

math.AP↗

arXiv2016

Well-posedness and long-time behavior for the Westervelt equation with absorbing boundary conditions of order zero

We investigate the Westervelt equation from nonlinear acoustics, subject to nonlinear absorbing boundary conditions of order zero, which were recently proposed by Kaltenbacher & Shevchenko. We apply the concept of maximal regularity of type $L_p$ to prove global well-posedness for small initial data. Moreover, we show that the solutions regularize instantaneously which means that they are $C^\infty$ with respect to time $t$ as soon as $t>0$. Finally, we show that each equilibrium is stable and each solution which starts sufficiently close to an equilibrium converges at an exponential rate to a possibly different equilibrium.

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