Interior derivative blow-up for quasilinear parabolic equations
Yoshikazu Giga
Discrete & Continuous Dynamical Systems - A 1995, 1(3): 449-461 doi: 10.3934/dcds.1995.1.449
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.
keywords: blow-up. quasilinear parabolic equations
$L^1$ maximal regularity for the laplacian and applications
Yoshikazu Giga Jürgen Saal
Conference Publications 2011, 2011(Special): 495-504 doi: 10.3934/proc.2011.2011.495
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
On behavior of signs for the heat equation and a diffusion method for data separation
Mi-Ho Giga Yoshikazu Giga Takeshi Ohtsuka Noriaki Umeda
Communications on Pure & Applied Analysis 2013, 12(5): 2277-2296 doi: 10.3934/cpaa.2013.12.2277
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.
keywords: equi-distance hypersurface Diffusive sign heat equations. sign-changing
A subdifferential interpretation of crystalline motion under nonuniform driving force
Mi-Ho Giga Yoshikazu Giga
Conference Publications 1998, 1998(Special): 276-287 doi: 10.3934/proc.1998.1998.276
Please refer to Full Text.
A counterexample to finite time stopping property for one-harmonic map flow
Yoshikazu Giga Hirotoshi Kuroda
Communications on Pure & Applied Analysis 2015, 14(1): 121-125 doi: 10.3934/cpaa.2015.14.121
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.
keywords: total variation flow. singular diffusion extinction One-harmonic map flow finite time stopping
Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations
Martino Bardi Yoshikazu Giga
Communications on Pure & Applied Analysis 2003, 2(4): 447-459 doi: 10.3934/cpaa.2003.2.447
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).
keywords: Hamilton-Jacobi equations differential games. semicontinuous viscosity solutions Cauchy problem right accessibility
Scale-invariant extinction time estimates for some singular diffusion equations
Yoshikazu Giga Robert V. Kohn
Discrete & Continuous Dynamical Systems - A 2011, 30(2): 509-535 doi: 10.3934/dcds.2011.30.509
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.
keywords: surface diffusion. Total variation flow extinction time
A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations
Yoshikazu Giga
Discrete & Continuous Dynamical Systems - S 2013, 6(5): 1277-1289 doi: 10.3934/dcdss.2013.6.1277
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.
keywords: Navier-Stokes equations Liouville problem non-slip boundary condition Poiseuille type flow.
Stability of facets of crystals growing from vapor
Yoshikazu Giga Piotr Rybka
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 689-706 doi: 10.3934/dcds.2006.14.689
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.
keywords: singular interfacial energy One-phase Stefan problem Gibbs-Thomson and kinetic effects stability of facets.
Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary
Yoshikazu Giga Przemysław Górka Piotr Rybka
Discrete & Continuous Dynamical Systems - A 2010, 26(2): 493-519 doi: 10.3934/dcds.2010.26.493
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.
keywords: comparison principle. singular energies discontinuous Hamiltonian Hamilton-Jacobi equation free boundary driven curvature flow

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