359 research outputs found
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
Helicoidal surfaces with constant anisotropic mean curvature
We study surfaces with constant anisotropic mean curvature which are
invariant under a helicoidal motion. For functionals with axially symmetric
Wulff shapes, we generalize the recently developed twizzler representation of
Perdomo to the anisotropic case and show how all helicoidal constant
anisotropic mean curvature surfaces can be obtained by quadratures
Universality in Blow-Up for Nonlinear Heat Equations
We consider the classical problem of the blowing-up of solutions of the
nonlinear heat equation. We show that there exist infinitely many profiles
around the blow-up point, and for each integer , we construct a set of
codimension in the space of initial data giving rise to solutions that
blow-up according to the given profile.Comment: 38 page
Very Singular Diffusion Equations-Second and Fourth Order Problems
This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example
Fine properties of self-similar solutions of the Navier-Stokes equations
We study the solutions of the nonstationary incompressible Navier--Stokes
equations in , , of self-similar form , obtained from small and homogeneous initial
data . We construct an explicit asymptotic formula relating the
self-similar profile of the velocity field to its corresponding initial
datum
Partial Regularity of solutions to the Four-dimensional Navier-Stokes equations at the first blow-up time
The solutions of incompressible Navier-Stokes equations in four spatial
dimensions are considered. We prove that the two-dimensional Hausdorff measure
of the set of singular points at the first blow-up time is equal to zero.Comment: 19 pages, a comment regarding five or higher dimensional case is
added in Remark 1.3. accepted by Comm. Math. Phy
Singular and regular solutions of a non-linear parabolic system
We study a dissipative nonlinear equation modelling certain features of the
Navier-Stokes equations. We prove that the evolution of radially symmetric
compactly supported initial data does not lead to singularities in dimensions
. For dimensions we present strong numerical evidence supporting
existence of blow-up solutions. Moreover, using the same techniques we
numerically confirm a conjecture of Lepin regarding existence of self-similar
singular solutions to a semi-linear heat equation.Comment: 16 page
Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains
We establish the resolvent estimates for the Stokes operator in
Lipschitz domains in , for . The result, in particular, implies that the Stokes operator in a
three-dimensional Lipschitz domain generates a bounded analytic semigroup in
for (3/2)-\varep < p< 3+\epsilon. This gives an affirmative answer to a
conjecture of M. Taylor.Comment: 28 page. Minor revision was made regarding the definition of the
Stokes operator in Lipschitz domain
Spikes and diffusion waves in one-dimensional model of chemotaxis
We consider the one-dimensional initial value problem for the viscous
transport equation with nonlocal velocity with a given kernel . We show the existence
of global-in-time nonnegative solutions and we study their large time
asymptotics. Depending on , we obtain either linear diffusion waves ({\it
i.e.}~the fundamental solution of the heat equation) or nonlinear diffusion
waves (the fundamental solution of the viscous Burgers equation) in asymptotic
expansions of solutions as . Moreover, for certain aggregation
kernels, we show a concentration of solution on an initial time interval, which
resemble a phenomenon of the spike creation, typical in chemotaxis models
Global existence of solutions to 2-D Navier-Stokes flow with non-decaying initial data in half-plane
We investigate the Navier-Stokes initial boundary value problem in the
half-plane with initial data
or with non decaying initial data . We introduce a technique that allows to solve the two-dimesional
problem, further, but not least, it can be also employed to obtain weak
solutions, as regards the non decaying initial data, to the three-dimensional
Navier-Stokes IBVP. This last result is the first of its kind
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