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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.
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 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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