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

主要研究人员小川教授取得了以下研究成果。他研究了临界型的Sobolev型不等式，特别是对实插值空间，如Besov和Triebel-Lirzorkin空间，并将其推广到抽象的Besov和Lorentz空间。涉及对数插值阶的问题可以应用于半线性偏微分方程解的正则性和唯一性准则。在与研究人员的一系列合作中，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。用类似的方法，给出了二维球面简谐热流光滑解的正则性判据。特别地，对于简谐热流的弱解，相似正则性准则也成立。结果是通过建立单调性公式…得到的进一步讨论了解的能量密度的平均振动性。他还考虑了非局部型半线性抛物型方程解的渐近行为。这些系统出现在不同的物理尺度上，如半导体模拟模型、趋化模型和天文学中恒星的诞生。该系统将泊松方程作为电荷或粘性阿米巴的密度所产生的场，非局域效应对解的分析是必不可少的。特别研究了临界情况下的二维情形，证明了强迫漂移扩散系统在临界Hardy空间中存在时间局部解，存在直到阈值初始密度的时间整体解，存在有限时间爆破。此外，对于小数据，解的非光滑行为用热核来刻画。研究了半线性阻尼波方程在全空间、半空间和外部区域的解的渐近性，证明了小的解可以分解为线性热方程的解，即具有非线性效应的线性波动方程的解.这一点以前已经在一维和三维情况下得到了证明，但这种方法不能适用于二维情况。较少

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