2,022 research outputs found
Anisotropic total variation flow of non-divergence type on a higher dimensional torus
We extend the theory of viscosity solutions to a class of very singular
nonlinear parabolic problems of non-divergence form in a periodic domain of an
arbitrary dimension with diffusion given by an anisotropic total variation
energy. We give a proof of a comparison principle, an outline of a proof of the
stability under approximation by regularized parabolic problems, and an
existence theorem for general continuous initial data, which extend the results
recently obtained by the authors.Comment: 27 page
Periodic total variation flow of non-divergence type in Rn
We introduce a new notion of viscosity solutions for a class of very singular
nonlinear parabolic problems of non-divergence form in a periodic domain of
arbitrary dimension, whose diffusion on flat parts with zero slope is so strong
that it becomes a nonlocal quantity. The problems include the classical total
variation flow and a motion of a surface by a crystalline mean curvature. We
establish a comparison principle, the stability under approximation by
regularized parabolic problems, and an existence theorem for general continuous
initial data.Comment: 36 pages, 2 figure
A caricature of a singular curvature flow in the plane
We study a singular parabolic equation of the total variation type in one
dimension. The problem is a simplification of the singular curvature flow. We
show existence and uniqueness of weak solutions. We also prove existence of
weak solutions to the semi-discretization of the problem as well as convergence
of the approximating sequences. The semi-discretization shows that facets must
form. For a class of initial data we are able to study in details the facet
formation and interactions and their asymptotic behavior. We notice that our
qualitative results may be interpreted with the help of a special composition
of multivalued operators
A level set crystalline mean curvature flow of surfaces
We introduce a new notion of viscosity solutions for the level set
formulation of the motion by crystalline mean curvature in three dimensions.
The solutions satisfy the comparison principle, stability with respect to an
approximation by regularized problems, and we also show the uniqueness and
existence of a level set flow for bounded crystals.Comment: 55 pages, 4 figure
Existence and uniqueness for a crystalline mean curvature flow
An existence and uniqueness result, up to fattening, for a class of
crystalline mean curvature flows with natural mobility is proved. The results
are valid in any dimension and for arbitrary, possibly unbounded, initial
closed sets. The comparison principle is obtained by means of a suitable weak
formulation of the flow, while the existence of a global-in-time solution
follows via a minimizing movements approach
Lorentz space estimates for vector fields with divergence and curl in Hardy spaces
In this note, we establish the estimate on the Lorentz space for
vector fields in bounded domains under the assumption that the normal or the
tangential component of the vector fields on the boundary vanishing. We prove
that the norm of the vector field can be controlled by the norms of
its divergence and curl in the atomic Hardy spaces and the norm of the
vector field itself.Comment: 11page
Energy solutions to one-dimensional singular parabolic problems with data are viscosity solutions
We study one-dimensional very singular parabolic equations with periodic
boundary conditions and initial data in , which is the energy space. We
show existence of solutions in this energy space and then we prove that they
are viscosity solutions in the sense of Giga-Giga.Comment: 15 page
Bent rectangles as viscosity solutions over a circle
We study the motion of the so-called bent rectangles by the singular weighted mean curvature. We are interested in the curves which can be rendered as graphs over a smooth onedimensional reference manifold. We establish a sufficient condition for that. Once we deal with graphs we can have the tools of the viscosity theory available, like the Comparison Principle. With its help we establish uniqueness of variational solutions constructed by the authors [18]. In addition, we establish a criterion for the mobility coefficient guaranteeing vertex preservation
Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data
In this paper, we prove the global well-posedness for the 3D rotating
Navier-Stokes equations in the critical functional framework. Especially, this
result allows to construct global solutions for a class of highly oscillating
initial data.Comment: 20page
Asymptotics of solutions to the Navier-Stokes system in exterior domains
We consider the incompressible Navier-Stokes equations with the Dirichlet
boundary condition in an exterior domain of with . We
compare the long-time behaviour of solutions to this initial-boundary value
problem with the long-time behaviour of solutions of the analogous Cauchy
problem in the whole space . We find that the long-time
asymptotics of solutions to both problems coincide either in the case of small
initial data in the weak -space or for a certain class of large initial
data
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