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### Open Access Journals

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.

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 consider the weighted mean curvature flow in the plane
with a driving term. For certain anisotropy functions this evolution problem
degenerates to a first order Hamilton-Jacobi equation with a free
boundary. The resulting problem may be written as a Hamilton-Jacobi
equation with a spatially non-local and discontinuous Hamiltonian. We
prove existence and uniqueness of solutions. On the way we show a
comparison principle and a stability theorem for viscosity solutions.

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