56 research outputs found
First-order evolution equations with dynamic boundary conditions: Dynamic boundary conditions
No full textIn this paper, we introduce a general framework to study linear first-order evolution equations on a Banach space X with dynamic boundary conditions, that is with boundary conditions containing time derivatives. Our method is based on the existence of an abstract Dirichlet operator and yields finally to equivalent systems of two simpler independent equations. In particular, we are led to an abstract Cauchy problem governed by an abstract Dirichlet-to-Neumann operator on the boundary space ∂X. Our approach is illustrated by several examples and various generalizations are indicated. This article is part of the theme issue 'Semigroup applications everywhere'
Operators with Wentzell boundary conditions and the Dirichlet‐to‐Neumann operator
No full textIn this paper we relate the generator property of an operator A with (abstract) generalized Wentzell boundary
conditions on a Banach space X and its associated (abstract) Dirichlet-to-Neumann operator N acting on a
“boundary” space ∂X. Our approach is based on similarity transformations and perturbation arguments and
allows to split A into an operator A_00 with Dirichlet-type boundary conditions on a space X 0 of states having
“zero trace” and the operator N . If A_00 generates an analytic semigroup, we obtain under a weak Hille–Yosida
type condition that A generates an analytic semigroup on X if and only if N does so on ∂X. Here we assume
that the (abstract) “trace” operator L : X → ∂X is bounded what is typically satisfied if X is a space of
continuous functions. Concrete applications are made to various second order differential operators
Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds
No full textWe consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space C(∂M) of continuous functions on the boundary ∂M of a compact manifold M with boundary. We prove that it generates an analytic semigroup of angle π/2, generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle π/2 on the space C(M)
Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds
No full textWe consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space C(∂M) of continuous functions on the boundary ∂M of a compact manifold M with boundary. We prove that it generates an analytic semigroup of angle π/2, generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle π/2 on the space C(M)
A convergent finite element algorithm for mean curvature flow in arbitrary codimension
No full textOptimal-order uniform-in-time H1-norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulas. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least 2 and backward difference formulas of orders 2 to 5. Numerical experiments in codimension 2 illustrate and complement our theoretical results
Global Wellposedness of the Primitive Equations with Nonlinear Equation of State in Critical Spaces
No full textThis article investigates the primitive equations with nonlinear equations of state. A global, strong well-posedness result for this set of equations is established for initial data lying in critical spaces provided that the density, depending on temperature, salinity and pressure, satisfies certain regularity assumptions. These assumptions are in particular satisfied for the TEOS-80 formulation of the equation of state
Global Wellposedness of the Primitive Equations with Nonlinear Equation of State in Critical Spaces
No full textThis article investigates the primitive equations with nonlinear equations of state. A global, strong well-posedness result for this set of equations is established for initial data lying in critical spaces provided that the density, depending on temperature, salinity and pressure, satisfies certain regularity assumptions. These assumptions are in particular satisfied for the TEOS-80 formulation of the equation of state
Rigorous analysis of the interaction problem of sea ice with a rigid body
No full textConsider the set of equations modeling the motion of a rigid body enclosed in
sea ice. Using Hibler's viscous-plastic model for describing sea ice, it is
shown by a certain decoupling approach that this system admits a unique, local
strong solution within the -setting.Comment: Accepted for publication in Mathematische Annale
Rigorous analysis of the interaction problem of sea ice with a rigid body
No full textConsider the set of equations modeling the motion of a rigid body enclosed in sea ice. Using Hibler’s viscous-plastic model for describing sea ice, it is shown by a certain decoupling approach that this system admits a unique, local strong solution within the Lp-setting
Interaction of the primitive equations with sea ice dynamics
No full textThis article establishes local strong well-posedness and global strong
well-posedness close to constant equilibria of a model coupling the primitive
equations of ocean and atmospheric dynamics with Hibler's viscous-plastic sea
ice model. In order to treat the coupling conditions, an approach involving the
hydrostatic Dirichlet and Dirichlet-to-Neumann operator is developed. Mapping
properties of the latter operators are investigated for the first time and are
of central importance for showing that the operator associated with the
linearized coupled system admits a bounded -calculus on
suitable -spaces. Quasilinear methods allow then to obtain the
strong well-posedeness results described above
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