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Preprint

Here is the list of available preprints (by mainly PDF file) with short descriptions.

T. Ogawa and S. Shimizu, Global well-posedness for the incompressible Navier-Stokes equations in the critical Besov space under the Lagrangian coordinates , J. Differential Equations 274 (2021), p. 613–651.

Abstract : We consider global well-posedness of the Cauchy problem of the incompressible Navier–Stokes equations under the Lagrangian coordinates in scaling critical Besov spaces. We prove the system is globally well-posed in the homogeneous Besov space $\dot{B}^{-1+n/p}_{p,1}(\mathbb{R}^n)$ with $1 \le p < \infty$. The former result was restricted for $1 \le p < 2n$ and the main reason why the well-posedness space is enlarged is that the quasi-linear part of the system has a special feature called a multiple divergence structure and the bilinear estimate for the nonlinear terms are improved by such a structure. Our result indicates that the Navier–Stokes equations can be transferred from the Eulerian coordinates to the Lagrangian coordinates even for the solution in the limiting critical Besov spaces.


T. Matsui, R. Nakasato and T. Ogawa, Singular limit for the magnetohydrodynamics of the damped wave type in the critical Fourier-Sobolev space , J. Differential Equations 271 (2021), p. 414–446.

Abstract : We study the Cauchy problem of the incompressible damped wave type magnetohydrodynamic system in $\mathbb{R}^N$ ($N \geq 2$). The purpose of this paper is to show the global well-posedness and a singular limit of the problem in Fourier–Sobolev spaces. For the proof of the results, we use the $L^p-L^q$ type estimates for the fundamental solutions of the damped wave equation and end-point maximal regularity for the inhomogeneous heat equation in that space with a detailed estimate of difference between the symbol of the heat kernel and fundamental solution of the damped wave equation.


T. Ogawa and S. Shimizu, Maximal $L^1$-regularity for parabolic boundary value problems with inhomogeneous data in the half-space , Proc. Japan Acad. Ser. A Math. Sci. 96 (2020), no. 7, p. 57–62.

Abstract : End-point maximal $L^1$-regularity for the parabolic initial-boundary value problem is considered in the half-space. For the inhomogeneous boundary data of both the Dirichlet and the Neumann type, maximal $L^1$-regularity for the initial-boundary value problem of parabolic equation is established in time end-point case upon the Besov space as well as the optimal trace estimates. We derive the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity.


M. Kurokiba and T. Ogawa, Singular limit problem for the two-dimensional Keller-Segel system in scaling critical space , J. Differential Equations 269 (2020), no. 10, p. 8959–8997.

Abstract : We consider the singular limit problem of the Cauchy problem to the Keller-Segel equation in the two dimensional critical space. It is shown that the solution to the Keller-Segel system in the scaling critical function space converges to the solution to the drift-diffusion system of parabolic-elliptic equations (the simplified Keller-Segel equation) in the critical space strongly as the relaxation time parameter . For the proof, we show generalized maximal regularity for the heat equations and use it systematically with the sequence of embeddings between the interpolation spaces $\dot{B}^{s}_{q,\sigma}(\mathbb{R}^2)$ and $\dot{F}^s_{q,\sigma}(\mathbb{R}^2)$ for the proof of singular limit problem.


M. Kurokiba and T. Ogawa, Singular limit problem for the Keller-Segel system and drift-diffusion system in scaling critical spaces , J. Evol. Equ. 20 (2020), no. 2, 421–457.

Abstract : We consider a singular limit problem for the Cauchy problem of the Keller–Segel equation in a critical function space. We show that a solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift–diffusion system of parabolic–elliptic type (the simplified Keller–Segel model) in the critical space strongly as the relaxation time $\tau \to \infty$. For the proof of singular limit problem, we employ generalized maximal regularity for the heat equation and use it systematically with the sequence of embeddings between the interpolation spaces $\dot{B}^s_{q,\sigma}(\mathbb{R}^n)$ and $\dot{F}^s_{q,\sigma}(\mathbb{R}^n)$.


T. Ogawa and T. Sato, Analytic smoothing effect for system of nonlinear Schrödinger equations with general mass resonance , Hiroshima Math. J. 50 (2020), no. 1, p. 73–84.

Abstract : We prove the analytic smoothing e¤ect for solutions to the system of nonlinear Schrödinger equations under the gauge invariant nonlinearities. This result extends the known result due to Hoshino [Nonlinear Differential Equations Appl. 24 (2017), Art. 62]. Under rapidly decaying condition on the initial data, the solution shows a smoothing effect and is real analytic with respect to the space variable. Our theorem covers not only the case for the gauge invariant setting but also multiple component case with higher power nonlinearity up to the fifth order.


T. Ogawa and T. Sato, $L^2$-decay rate for the critical nonlinear Schrödinger equation with a small smooth data , NoDEA Nonlinear Differential Equations Appl. 27 (2020), no. 2, Paper No. 18, 20 pp.

Abstract : We consider the Cauchy problem for the one dimensional nonlinear dissipative Schrödinger equation with a cubic nonlinearity $\lambda |u|^2u$, where $\lambda \in \mathbb{C}$ with Im $\lambda < 0$. We show that a relation between $L^2$-decay rate for the solution and a smoothness of the initial data. Our result improves the recent work of Hayashi–Li–Naumkin (Adv Math Phys Art. ID 3702738, 7, 2016) for the decay rate of $L^2$.


N. Hayashi, E. I. Kaikina and T. Ogawa, Dirichlet-boundary value problem for one dimensional nonlinear Schrödinger equations with large initial and boundary data , NoDEA Nonlinear Differential Equations Appl. 27 (2020), no. 2, Paper No. 17, 20 pp.

Abstract : We consider the inhomogeneous Dirichlet-boundary value problem with large initial and boundary data for nonlinear Schrödinger equations in one space dimension. Global existence and asymptotic behavior in time of solutions to the problem are obtained by using the classical energy method and factorization techniques.


M.R. Haque, T. Ogawa and R. Sato, Existence of weak solutions to a convection-diffusion equation in a uniformly local Lebesgue space , Commun. Pure Appl. Anal. 19 (2020), p. 677–697.

Abstract : We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection–diffusion equation in a uniformly local function space $L^r_{uloc,p}(\Omega)$, where the solution is not decaying at $|x| \to \infty$. We show that the local existence and the uniqueness of a solution for the initial data in uniformly local $L^r$ spaces and identify the Fujita-Weissler critical exponent for the local well-posedness found by Escobedo-Zuazua [10] is also valid for the uniformly local function class.


T. Ogawa and H. Wakui, inite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition , Manuscripta Math. 159 (2019), p. 475–509.

Abstract : We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initial data has negative free energy, then either the corresponding weak solution to the equation does not exist globally in time, or the time global solution does not remain bounded in the energy space. We emphasize that our result does not require any weight assumption on the initial data, and hence, a solution may have an infinite second moment. The proof is based upon the modified virial law and conservation laws and we show that the modified moment functional vanishes for a finite time under the negative energy condition. For a radially symmetric case, the solution blows up in finite time and the mass concentration phenomenon occurs with a sharp lower bound related to the best constant for the Hardy–Littlewood–Sobolev inequality.


M. Kurokiba and T. Ogawa, Finite time blow up for solutions to a degenerate drift-diffusion equation for a fast diffusion case , Nonlinearity 32 (2019), p. 2073-2093.

Abstract : We consider the non-existence of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the fast diffusion exponent. We show the solution for the fast diffusion cases with the diffusion exponent $\frac{n}{n+2} < \alpha < 1$ blows up in a finite time if the initial data satisfies certain condition involving the free energy. We also show the finite time blow up for radially symmetric case without finite moment condition.


N. Ioku and T. Ogawa, Critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations , J. Differential Equations 266 (2019), p. 2274--2293

Abstract : We consider a heat semigroup with an inverse square potential and prove critical dissipative estimates including the endpoint case. Some application to a nonlinear heat equation with an inverse square potential is also discussed.


H. Kubo, T. Ogawa, and T. Suguro, Beckner type of the logarithmic Sobolev and a new type of Shannon's inequalities and an application to the uncertainty principle , Proc. Amer. Math. Soc. 147 (2019), pp. 1511--1518.

Abstract : We consider the inequality which has a "dual" relation with Beckner's logarithmic Sobolev inequality. By using the relative entropy, we identify the sharp constant and the extremal of this inequality. Moreover, we derive the logarithmic uncertainty principle like Beckner's one.


H. Wakui, The rate of concentration for the radially symmetric solution to a degenerate drift-diffusion equation with the mass critical exponent , Arch. Math. 111 (2018), p. 535 -- 548

Abstract : We consider the concentration rate of the total mass for radially symmetric blow-up solutions to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. We proved that the radially symmetric solution blows up in finite time when the initial data has negative free energy. We show that the mass concentration phenomenon occurs with the sharp lower constant related to the best constant of the Hardy-Littlewood-Sobolev inequality and the concentration rate of the total mass.


H. Wakui, Unboundedness for solutions to a degenerate drift-diffusion equation with the $L^1$-supercritical and the energy subcritical exponent , J. Math. Anal. Appl. 461 (2018), p.1686--1710

Abstract : We consider large time behavior of weak solutions to a degenerate drift-diffusion system related to Keller-Segel system with the $L^1$-supercritical and the energy subcritical cases under relaxed weight condition. It is known that the large time behavior of solutions is classified by the invariant norms of initial data. For the $L^1$-critical case, Ogawa-Wakui proved that the negative entropy condition induces the unboundedness of corresponding solutions with the initial data decaying slowly at spacial infinity. Here the result is a continuous analogy of the $L^1$-critical case. Analogous results have been obtained in the theory of nonlinear Schrödinger equations.


T. Ogawa and H. Wakui, Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition , Accepted for publication in manuscripta mathematica , DOI : https://doi.org/10.1007/s00229-019-01108-x

Abstract : We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initial data has a negative free energy, then the corresponding weak solution to the equation does not exist globally in time or the time global solution does not remain bounded for the scaling critical space. We emphasize that our result does not require any weight assumption on the initial data so that a solution with infinite second moment may be allowed. The proof is based on the modified virial law and observing that the modified moment functional vanishes for some time. For a radially symmetric case, the solution blows up in a finite time and the mass concentration phenomenon occurs with the sharp lower constant related to the best constant for the Hardy-Littlewood-Sobolev inequality.


S. Machihara and T. Ogawa, Global well-posedness for one dimensional Chern-imon-Dirac system in $L^p$ , Comm. Partial Differential Equations 42 (2017), p.1175--1198

Abstract : Time global wellposedness in $L^p$ for the Chern-Simons-Dirac equation in $1 + 1$ dimension is discussed. The two types of quadratic terms are treated, null case and non-null case. The standard iteration arguments, but settings correspond to each cases respectively, are used for the proof. For the critical case in $L^1$, the mass concentration phenomena of the solutions is denied to show the time global solvability. The intrinsic estimate plays an important role in the proof. These arguements follow the work of Candy which showed the time global wellposedness for the single Dirac equation with cubic nonlinearity in critical $L^2$.


T. Ogawa and H. Wakui, Stability and Instability of Solutions to Drift-diffusion Equations , Evol. Equ. Control Theory 6 (2017), p.587--597

Abstract : We consider the large time behavior of a solution to a drift- diffusion equation for degenerate and non-degenerate type. We show an instability and uniform unbounded estimate for the semi-linear case and uniform bound and convergence to the stationary solution for the case of mass critical degenerate case for higher space dimensions than two.


T. Ogawa and Y. Yamane, Well-posedness of the compressible Navier-Stokes-Poisson system in the critical Besov spaces , Springer Proc. Math. Stat. 215 (2017), p.215--239

Abstract : We show the local well-posedness of the Cauchy problem to a nonlinear heat equation of Fujita type in lower space dimensions. It is well known that the non-negative solution corresponding to the Fujita critical exponent $p = 1 +\frac{2}{n}$ does not exist in the critical scaling invariant space $L^1(\mathbb{R}^n)$. We show if the initial data is in a modified Besov spaces, then the corresponding mild solution to the equation with the Fujita critical exponent $p = 1 + 2/n$ exists and the problem is local well-posed in the same space of the initial data. Besides we also show the problem is ill-posed in the scaling invariant Besov and inhomogeneous Besov spaces. This is known in $L^1$ space and extension of the result known in the Lebesgue spaces.


N. Chikami and T. Ogawa, Well-posedness of the compressible Navier-Stokes-Poisson system in the critical Besov spaces , J. Evol. Equ. 17 (2017), p.717--747

Abstract : We consider the Cauchy problem of the compressible Navier-Stokes system coupled with a Poisson equation. We give the optimal well-posedness in terms of scaling in the Besov framework. The results include the case of two dimensions, which is not treated in previous results.


T. Ogawa and H. Wakui, Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions , Anal. Appl. (Singap.) 14 (2016), p.145--183

Abstract : We show the non-uniform bound for a solution to the Cauchy problem of a drift–diffusion equation of a parabolic-elliptic type in higher space dimensions. If an initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not remain uniformly bounded in a scaling critical space. In other words, the solution grows up at $t \to \infty$ in the critical space or blows up in a finite time. Our presenting results correspond to the finite time blowing up result for the two-dimensional case. The proof relies on the logarithmic entropy functional and a generalized version of the Shannon inequality. We also give the sharp constant of the Shannon inequality.


T. Ogawa and S. Shimizu, End-point maximal $L^1$ regularity for the Cauchy problem to a parabolic equation with variable coeffcients , Math. Ann. 365 (2016), p.661--705

Abstract : We consider maximal $L^1$-regularity for the Cauchy problem to a parabolic equation in the Besov space $\dot{B}^0_{p,1}$ with $1 \leq p \leq \infty$. The estimate obtained here is not available by abstract theory of the class of unconditional martingale differences, because the end-point Besov space is included. We consider the end-point estimate and show that the optimality of maximal regularity in $L^1$ for the linear parabolic equation with variable coefficients.


S. Kawashima and Y. Ueda, Mathematical entropy and Euler-Cattaneo-Maxwell system , Anal. Appl. (SIngap.) 14 (2016), p.201--143

Abstract : In this paper, we introduce a notion of the mathematical entropy for hy- perbolic systems of balance laws with (not necessarily symmetric) relaxation. As applications, we deal with the Timoshenko system, the Euler-Maxwell system and the Euler-Cattaneo-Maxwell system. Especially, for the Euler-Cattaneo-Maxwell system, we observe that its dissipative structure is of the regularity-loss type and investigate the corresponding decay property. Furthermore, we prove the global existence and asymptotic stability of solutions to the Euler-Cattaneo-Maxwell system for small initial data.