arXiv subjects
Publications and source records attributed to Hideyuki Miura.
We consider the semilinear heat equation $u_t=Δu+|u|^{p-1} u$ in possibly non-convex and unbounded domains. Our main result shows the nonexistence of type II blow-up for possibly sign-changing solutions in the energy subcritical range $(n-2)p<n+2$. This resolves a long-standing open question dating back to the 1980s and also deduces the blow-up of the scaling critical norm.
We consider the SQG equation without dissipation on the half-plane with Dirichlet boundary condition, and prove local wellposedness in the spaces $W^{3,p}$ and $C^{2,β}$ for any $1<p<\infty$ and $0<β<1$. We complement this wellposedness by showing that for generic $C^{\infty}_{0}$ initial data, the unique corresponding solution does not belong to $W^{3,\infty}$.
We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded $L^{n(p-1)/2,\infty}(\mathbf{R}^n)$-norm up to the blow-up time. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative $\varepsilon$-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.
We address the critical norm blow-up problem for the nonlinear heat equation $u_t-Δu=|u|^{p-1}u$ in $\mathbf{R}^n\times(0,T)$. In the supercritical range $p>(n+2)/(n-2)$, we prove that if the maximal existence time $T$ is finite, then $\lim_{t\to T}\|u(\cdot,t)\|_{L^{n(p-1)/2}(\mathbf{R}^n)} =\infty$ without assuming extra conditions such as radial symmetry or the type of blow-up.
We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=Δu+|u|^{p-1}u$ in an arbitrary smooth domain of $\mathbf{R}^n$. In the range $p>p_S:=(n+2)/(n-2)$, we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range $p>p_S$ is optimal in view of the existence of type II blow-up solutions with bounded critical norm for $p=p_S$.
We study the regular sets of local energy solutions to the Navier-Stokes equations in terms of conditions on the initial data. It is shown that if a weighted $L^2$ norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math. 35; 1982) and our recent paper [15] as well.
We prove an $ε$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular in a set enclosed by a paraboloid started from the origin. The result is applied to the estimate of the regular set for local energy solutions with initial data in weighted $L^2$ spaces. We also apply this result to studying energy concentration near a possible blow-up time and regularity of forward discretely self-similar solutions.
We study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this paper it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondary blow-up is also observed when the strain rate is relatively weak.
This paper is devoted to the study of the Stokes and Navier-Stokes equations, in a half-space, for initial data in a class of locally uniform Lebesgue integrable functions, namely $L^q_{uloc,σ}(\R^d_+)$. We prove the analyticity of the Stokes semigroup $e^{-t{\bf A}}$ in $L^q_{uloc,σ}(\R^d_+)$ for $1<q\leq\infty$. This follows from the analysis of the Stokes resolvent problem for data in $L^q_{uloc,σ}(\R^d_+)$, $1<q\leq\infty$. We then prove bilinear estimates for the Oseen kernel, which enables to prove the existence of mild solutions. The three main original aspects of our contribution are: (i) the proof of Liouville theorems for the resolvent problem and the time dependent Stokes system under weak integrability conditions, (ii) the proof of pressure estimates in the half-space and (iii) the proof of a concentration result for blow-up solutions of the Navier-Stokes equations. This concentration result improves a recent result by Li, Ozawa and Wang and provides a new proof.
The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space $\mathbb R^3_+$. Such solutions are sometimes called Lemarié-Rieusset solutions in the whole space $\mathbb R^3$. The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz-Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical $L^3(\mathbb R^3_+)$ norm obtained by Barker and Seregin for solutions developing a singularity in finite time.
We prove short time regularity of suitable weak solutions of 3D incompressible Navier-Stokes equations near a point where the initial data is locally in $L^3$. The result is applied to the regularity problems of solutions with uniformly small local $L^3$ norms, and of forward discretely self-similar solutions.
We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved.
We derive refined estimates of the Green tensor of the stationary Stokes system in the half space. We then investigate the spatial asymptotics of stationary solutions of the incompressible Navier-Stokes equations in the half space. We also discuss the asymptotics of fast decaying flows in the whole space and exterior domains. In the Appendix we consider axisymmetric self-similar solutions.
We consider the two-dimensional Euler equations in non-smooth domains with corners. It is shown that if the angle of the corner $θ$ is strictly less than $π/2$, the Lipschitz estimate of the vorticity at the corner is at most single exponential growth and the upper bound is sharp. %near the stagnation point. For the corner with the larger angle $π/2 < θ<2π$, $θ\neq π$, we construct an example of the vorticity which loses continuity instantaneously. For the case $θ\le π/2$, the vorticity remains continuous inside the domain. We thus identify the threshold of the angle for the vorticity maintaining the continuity. For the borderline angle $θ=π/2$, it is also shown that the growth rate of the Lipschitz constant of the vorticity can be double exponential, which is the same as in Kiselev-Sverak's result (Annals of Math., 2014).
In this paper, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_1,x_2):0<x_1+x_2<\sqrt{2},0<-x_1+x_2<\sqrt{2}\}$ is considered. It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.
Let $Ω$ be a domain with noncompact boundary. It is known that the Helmholtz decomposition is not always valid in $L^p(Ω)$ except for the energy space $L^2 (Ω)$. In this paper we consider a typical unbounded domain whose boundary is given as a Lipschitz graph, and show that the Helmholtz decomposition holds in certain anisotropic spaces which include some infinite energy vector fields.
We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators related with Poisson operators and Dirichlet-Neumann maps. Consequently, we obtain a solution formula for the inhomogeneous elliptic boundary value problem in the half space, which is useful to show the existence of solutions in a wider class of inhomogeneous data. We also establish $L^2$ solvability of boundary value problems for a new class of the elliptic operators.