Papers

Preprint

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

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M. Kurokiba and T. Ogawa, Finite time blow up for solutions to a degenerate drift-diffusion equation for a fast diffusion case , Accepted for publication in Nonlinearity

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.

(30 pages)

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 $p = 1 + 2/n$ 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.