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PROC

Inter alia we prove $L^1$ maximal regularity for the Laplacian in the
space of Fourier transformed nite Radon measures FM. This is remarkable,
since FM is not a UMD space and by the fact that we obtain $L_p$ maximal
regularity for $p$ = 1, which is not even true for the Laplacian in $L^2$. We apply
our result in order to construct strong solutions to the Navier-Stokes equations
for initial data in FM in a rotating frame. In particular, the obtained results
are uniform in the angular velocity of rotation.

keywords:
Radon measures
,
strong solutions
,
Navier-Stokes equations
,
Coriolis
force
,
Maximal regularity

CPAA

Consider the solution $u(x,t)$ of the heat equation
with initial data $u_0$.
The diffusive sign $S_D[u_0](x)$ is defined
by the limit of sign of $u(x,t)$ as $t\to 0$.
A sufficient condition for $x\in R^d$
and $u_0$ such that $S_D[u_0](x)$ is well-defined is given.
A few examples of $u_0$ violating and fulfilling
this condition are given.
It turns out that this diffusive sign
is also related to variational problem
whose energy is the Dirichlet energy
with a fidelity term.
If initial data is a difference of characteristic functions
of two disjoint sets, it turns out that the boundary
of the set $S_D[u_0](x) = 1$ (or $-1$) is roughly
an equi-distance hypersurface from $A$ and $B$ and
this gives a separation of two data sets.

PROC

Please refer to Full Text.

keywords:

CPAA

For a very strong diffusion equation like total variation flow it is often observed that the solution stops at a steady state in a finite time. This phenomenon is called a finite time stopping or a finite time extinction if the steady state is zero. Such a phenomenon is also observed in one-harmonic map flow from an interval to a unit circle when initial data is piecewise constant. However, if the target manifold is a unit two-dimensional sphere, the finite time stopping may not occur. An explicit example is given in this paper.

CPAA

We study Hamilton-Jacobi equations with upper semicontinuous initial data
without convexity assumptions on the Hamiltonian. We analyse the
behavior of generalized

*u.s.c*solutions at the initial time $t=0$, and find necessary and sufficient conditions on the Hamiltonian such that the solution attains the initial data along a sequence (right accessibility).
DCDS

We study three singular parabolic evolutions: the second-order total variation flow, the
fourth-order total variation flow, and a fourth-order surface diffusion law. Each has the property
that the solution becomes identically zero in finite time. We prove scale-invariant estimates for
the extinction time, using a simple argument which combines an energy estimate with a suitable
Sobolev-type inequality.

DCDS-S

We construct a Poiseuille type flow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included.

DCDS

Consider a Stefan-like problem with Gibbs-Thomson and kinetic
effects as a model of crystal growth from vapor. The equilibrium shape is
assumed to be a regular circular cylinder. Our main concern is a problem
whether or not a surface of cylindrical crystals (called a facet) is stable under evolution
in the sense that its normal velocity is constant over the facet. If a facet
is unstable, then it breaks or bends. A typical result we establish is that all
facets are stable if the evolving crystal is near the equilibrium. The
stability criterion we use is a variational principle for selecting the
correct Cahn-Hoffman vector. The analysis of the phase plane of an evolving
cylinder (identified with points in the plane) near the unique equilibrium
provides a bound for ratio of velocities of top and lateral facets of the
cylinders.

DCDS

We give examples of a bounded solution whose gradient
blows up in a finite time but it stays bounded on the boundary for a class of
quasilinear parabolic equations with zero boundary data.
The method reflects a geometric argument for curve evolution equations.

DCDS

In this paper we prove instant extinction of the solutions to Dirichlet and
Neumann boundary value problem for some quasilinear parabolic equations whose
diffusion coefficient is singular when the spatial gradient of unknown function is zero.

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