Kato Type Research Articles

Inviscid limit for stochastic Navier-Stokes equations under general initial conditions

We consider in a smooth and bounded two dimensional domain the convergence in the L2 norm, uniformly in time, of the solution of the stochastic Navier-Stokes equations with additive noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, under general regularity on the initial conditions of the Euler equations, that assuming the dissipation of the energy of the solution of the Navier-Stokes equations in a Kato type boundary layer, then the inviscid limit holds.

On Trotter–Kato type inductive limits in the category of C0-semigroups

On Trotter–Kato type inductive limits in the category of C0-semigroups

Global Well-Posedness for the Cauchy Problems of the 3d Simplified Ericksen–Leslie System with Fujita–Kato Type Initial Data

Global Well-Posedness for the Cauchy Problems of the 3d Simplified Ericksen–Leslie System with Fujita–Kato Type Initial Data

Euler System with a Polytropic Equation of State as a Vanishing Viscosity Limit

We consider the Euler system of gas dynamics endowed with the incomplete equation of state relating the internal energy to the mass density and the pressure. We show that any sufficiently smooth solution can be recovered as a vanishing viscosity - heat conductivity limit of the Navier--Stokes--Fourier system with a properly defined temperature. The result is unconditional in the case of the Navier type (slip) boundary conditions and extends to the no-slip condition for the velocity under some extra hypotheses of Kato's type concerning the behavior of the fluid in the boundary layer.

Global Wellposedness and Large Time Behavior of Solutions to the Hall-Magnetohydrodynamics Equations

We prove the global solutions to the incompressible Hall-magnetohydrodynamics (Hall-MHD) equations with small or some large initial data in \mathbb R^n (n \geq 3) . Especially, Fujita–Kato type initial data as the incompressible Navier–Stokes equations are allowed. We also study the large time behavior of the solutions and obtain an optimal decay rate in the general Besov spaces. Different from all the previous wellposedness results in [6, 7, 21], the initial energy we considered here may not be finite.

Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

In this work we consider several semilinear damped wave equations with “subcritical” nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between “wave-like” and “heat-like” behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the “transition surfaces” separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results.We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation.The main tool employed in the proof is a Kato's type lemma, established by iteration argument.

Harmonic 2-forms and positively curved 4-manifolds

We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a Kato type inequality, then it is definite. We also discuss some new insights for compact Riemannian 4-manifolds of positive sectional curvature.

Global Kato Type Smoothing Estimates via Local Ones for Dispersive Equations

In this paper we show that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schrodinger equation. From this we immediately have two results as follows. One is that the known local Kato smoothing estimates are sharp. The sharp regularity ranges of the global Kato smoothing estimates are already known, but those of the local Kato smoothing estimates are not. Sun et al. (Proc Am Math Soc 145(2):653–664, 2017) have shown it only in spacetime $$\mathbb R \times \mathbb R$$. Our result resolves this issue in higher dimensions. The other one is the sharp global-in-time maximal Schrodinger estimates. Recently, the pointwise convergence conjecture of the Schrodinger equation has been settled by Du et al. (Ann Math 186:607–640, 2017) and Du and Zhang (Ann Math 189:837–861, 2019). For this they proved related sharp local-in-time maximal Schrodinger estimates. By our result, these lead to the sharp global-in-time maximal Schrodinger estimates.

The Keller--Segel System on the Two-Dimensional-Hyperbolic Space

In this paper, we shall study the parabolic-elliptic Keller--Segel system on the Poincaré disk model of the two-dimensional-hyperbolic space. We shall investigate how the negative curvature of this Riemannian manifold influences the solutions of this system. As in the two-dimensional Euclidean case, under the subcritical condition $\chi M< 8\pi$, we shall prove global well-posedness results with any initial $L^1$-data. More precisely, by using dispersive and smoothing estimates we shall prove Fujita--Kato type theorems for local well-posedness. We shall then use the logarithmic Hardy--Littlewood--Sobolev estimates on the hyperbolic space to prove that the solution cannot blow up in finite time. For larger mass $\chi M> 8\pi$ we shall obtain a blow-up result under an additional condition. According to the exponential growth of the hyperbolic space, we find a suitable weighted moment of exponential type on the initial data for blow-up.

New Decompositions for Classes of Operators with Topological Uniform Descent

The article describes a new decomposition property for operators with topological uniform descent, like Kato type operators, as well as new results on the stability of this class of operators under perturbations by operators with finite-range power based on topological descent notion, from which we can generalize many perturbation results for a large classes of operators by extending to Banach spaces known techniques on Hilbert spaces. As application of our resuts we obtain that is a lower semi B-Weyl operator if and only if , where is a lower semi B-Browder operator and , for some . Our methods generalize to Banach spaces some results obtained by Aiena for operators acting on Hilbert spaces.

Multiplicity and Concentration of Solutions for a Fractional Kirchhoff Equation with Magnetic Field and Critical Growth

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points.

Global Well-Posedness for the 3D Incompressible Hall-Magnetohydrodynamic Equations with Fujita–Kato Type Initial Data

Hall-magnetohydrodynamic (Hall-MHD) equations which can be derived from two fluids model or kinetic models [see Acheritogaray et al. (Kinet Relat Models 4:901–918, 2011)] plays a crucial role in the study of magnetic reconnection in space plasmas, star formation, neutron stars. In this paper, we obtain two Fujita–Kato type results for the 3D Hall-MHD equations, which almost give positive answers to the question proposed by Chae and Lee (Remark 2 in Chae and Lee [J Differ Equ 256:3835–3858, 2014)]. The coupling between u and B is the main difficulty. Our idea is splitting the Navier–Stokes equations from the Hall-MHD equations and combining with some suitable blow-up criteria.

Kato's type theorems for the convergence of Euler-Voigt equations to Euler equations with Drichlet boundary conditions

After investigating existence and uniqueness of the global strong solutions for Euler-Voigt equations under Dirichlet conditions, we obtain the Kato's type theorems for the convergence of the Euler-Voigt equations to Euler equations. More precisely, the necessary and sufficient conditions that the solution of Euler-Voigt equation converges to the one of Euler equations, as $ \alpha\to 0 $, can be obtained.

New decompositions for the classes of quasi-Fredholm and semi B-Weyl operators

ABSTRACTIn this paper, we show that if is a quasi-Fredholm operator of degree d such that is complemented, then T has a decomposition like the Kato type operators. Using this decomposition allows us get a result about the stability of this class of operators under perturbations by nilpotent operators. Also we give a new decomposition for the class of semi B-Weyl operators, and through this property we shown that T is a semi B-Weyl operator if and only if T=S+K where S is a semi B-Browder operator and K is finite-dimensional.

Global well-posedness and exponential decay for fifth-order Korteweg–de Vries equation posed on the finite domain

This paper studies the fifth-order Korteweg–de Veries equation posed on finite interval(0.1){∂tu−∂x5u+ε1∂x4u+κ1∂x3u−ε2∂x2u+κ2∂xu+ε3u+u∂xu=0,x∈(0,1),t≥0,u(x,0)=φ(x),x∈(0,1),u(0,t)=u(1,t)=∂xu(0,t)=∂xu(1,t)=0,∂x2u(1,t)=ξ∂x2u(0,t),t≥0. Firstly, we establish smoothing effect of Kato type for the fifth-order Korteweg–de Veries equation posed on finite interval; Secondly, we show that the closed loop system (0.1) is globally well-posed in the space Hs(0,1) for s∈[0,5]; Lastly, we prove that the solution of the closed loop system (0.1) is exponentially decay in the space Hs(0,1) for s∈[0,5].

Decay estimates and Strichartz estimates of fourth-order Schrödinger operator

We study time decay estimates of the fourth-order Schrödinger operator H=(−Δ)2+V(x) in Rd for d=3 and d≥5. We analyze the low energy and high energy behaviour of resolvent R(H;z), and then derive the Jensen–Kato dispersion decay estimate and local decay estimate for e−itHPac under suitable spectrum assumptions of H. Based on Jensen–Kato type decay estimate and local decay estimate, we obtain the L1→L∞ estimate of e−itHPac in 3-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of e−itHPac for d≥5. Furthermore, using the local decay estimate and the Georgescu–Larenas–Soffer conjugate operator method, we prove the Jensen–Kato type decay estimates for some functions of H.

Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type

Semidiscretization in time is studied for a class of quasi-linear evolution equations in a framework due to Kato, which applies to symmetric first-order hyperbolic systems and to a variety of fluid and wave equations. In the regime where the solution is sufficiently regular, we show stability and optimal-order convergence of the linearly implicit and fully implicit midpoint rules and of higher-order implicit Runge–Kutta methods that are algebraically stable and coercive, such as the collocation methods at Gauss nodes.

Markov processes on the Lipschitz boundary for the Neumann and Robin problems

We investigate the Markov process on the boundary of a bounded Lipschitz domain associated to the Neumann and Robin boundary value problems. We first construct Lp-semigroups of sub-Markovian contractions on the boundary, generated by the boundary conditions, and we show that they are induced by the transition functions of the forthcoming processes. As in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected Brownian motion. The Robin problem is treated with a Kato type Lp-perturbation method, using the Revuz correspondence. An exceptional (polar) set occurs on the boundary. We make the link with the Dirichlet forms approach.

Generalized Kato Decomposition and Essential Spectra

Let R denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator T on a Banach space X we prove that $$T=T_M\oplus T_N$$ with $$T_M \in $$ R and $$T_N$$ quasinilpotent (nilpotent) if and only if T admits a generalized Kato decomposition (T is of Kato type) and 0 is not an interior point of the corresponding spectrum $$\sigma _\mathbf{R}(T) =\{\lambda \in \mathbb {C}: T-\lambda \notin \mathbf{R}\}$$ . Moreover, we prove that if $$T-\lambda _0$$ admits a generalized Kato decomposition, then $$\sigma _\mathbf{R}(T)$$ does not cluster at $$\lambda _0$$ if and only if $$\lambda _0$$ is not an interior point of $$\sigma _\mathbf{R}(T)$$ . As a consequence we get several results on cluster points of essential spectra. In that way we extend some results regarding the approximate point spectrum and the surjective spectrum given by Aiena and Rosas (J. Math. Anal. Appl. 279:180–188, 2003), as well as results given by Jiang and Zhong (J. Math. Anal. Appl. 356:322–327, 2009) to the cases of essential spectra.

Reduced measures for semilinear elliptic equations involving Dirichlet operators

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where $A=\Delta$ and show basic properties of solutions of (E). We also prove Kato's type inequality. Finally, we characterize the set of good measures in case $f(u)=-u^p$ for some $p>1$.