Reseach for the singularities and regularity of solutions to crtical nonlinear partial differential equations

主要研究者小川教授获得了以下结果。他研究了临界型Sobolev型不等式，特别是对真实的插值空间如Besov和Triebel-Lirzorkin空间的不等式，并将其推广到抽象的Besov和Lorentz空间。这些涉及对数插值阶的不等式可用于半线性偏微分方程的正则性和唯一性的判别。在一系列的合作研究中，他证明了三维Navier-Stokes方程弱解的正则性和唯一性准则以及Euer方程的破裂条件。在一个类似的方法，他还表明了规律性标准的顺利解决方案的二维谐波热流到一个球。特别地，对于调和热流的弱解，同样也成立了类似的正则性判据。结果是通过建立“单调性公式”得到的 ...更多信息 讨论了非局部型半线性抛物方程解的能量密度的平均振动性，并讨论了非局部型半线性抛物方程解的渐近性态。这些系统出现在各种物理标度中，如半导体模拟模型、趋化性模型和天文学中的星星诞生。该系统涉及泊松方程作为由电荷密度或粘液变形虫产生的场，非局部效应对于解的分析是必不可少的。特别是对临界情形，二维情形，证明了在临界哈代空间存在时间局部解，对于强迫漂移扩散情形，在达到阈值初始密度时存在时间整体解，并且存在有限时间爆破。此外，小数据下解的渐近性态可用热核来表征。此外，若场方程以某种非线性方式扰动，则在径向对称情形下，对于相同的初值，存在两个解.他还研究了半线性阻尼波动方程解在全空间、半空间和外部区域中的渐近性态，证明了小解将分解为线性热方程的解，具有非线性效应线性波动方程的某种组合。这在一维和三维情况下已经得到了证明，但这种方法不适用于二维情况。少

The main researcher, Prof.Ogawa obtained the following results. He researched for the Sobolev type inequality of the critical type, especially for the real interpolation spaces such as Besov and Triebel-Lirzorkin spaces and generalized it for the abstract Besov and Lorentz space. Those inqualities involving the logarithmic interpolation order can be applied for the regularity and uniqueness criterion of the seimilinear partial differential equation. In a series of collaboration with the research colabolators, he shows that the reguarlity and uniquness criterion for the weak solution of the 3 dimensional Navier-Stokes equations and break down condition for the Euer equation. In a similar method, he also showed the regularity criterion for the smooth solution of the 2 dimensional harmonic heat flow into a sphere. In particular, for the weak solution of the harmonic heat flow, the similar regularity criterion is also holds. The result is obtained by establishing the "monotonicity formula" … More for the mean oscillation of the energy density of the solutions.He also consider the asymptotic behavior of the solution for the semi-lineear parabolic equation of the non-local type. Those system appeared in a various Physical scaling such as semi-conductor simulation model, Chemotaxis model and the birth of star in Astronomy. The system is involving Poisson equation as the field generated by the dencity of the charge or mucous ameba and the non-local effect is essential for the analysis of the solution. He particulariy investigated to the critical situation, 2-dimensional case, and showed that there exists a time local solution in the critical Hardy space, time global solution upto the threshold initial density and finite time blow-up for the system of forcusing drift-diffusion case. Besides, the asymootitic behavior of the solution for small data is characterized by the heat kernel. Moreover if the field equation is purterbed in a certain nonlinear way, then there exist two solutions for the same initial data in a radially symmetric case.He also studied for the asymptotic behavior of the solution for the semi-linear damped wave equation in whole and half spaces and exterior domains and show the small solution is going to be decomposed into the solutions of the linear heat equation, some combination of linear wave equation with nonlinear effect. This was shown for 1 and 3 dimensional cases before, however the mothod there could not be applicable for the 2dimensional case. Less