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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.
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.
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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.
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).
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.
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.
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.
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.
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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