Research of Nonlinear Partial Differential Equations Using Variational Methods and Time-discretization Schemes (1999)

本研究的目标是研究非线性部分差异相等的解决方案，其中它们与使用变量的计算和时间离散方案的变量密切相关。更确切地说，对于某些可变函数F（u）我们处理部分差异方程D 7 μ u（/）μ t D 7-（欧拉-语言方程F） = 0。为了构建较弱的解决方案，以满足我们正在采取的行动。We consider the functional G-D2n-D2（u） =-D7| u-u-D2n-D2-1|（±）2 hイエD7dx+F（u）， and de\uイD2nイエD2 as a minimizer of GイD2nイエD2（u） successively。Combining {u-D2n-D2} by line segments we construct a approximate solution u-D2h-D2（x， t）。最后，在某些条件下，我们提出了你的D2 h D2（x， t）转换成一个较弱的解决方案。关于另一方，确定地图序列的限制{ui-D2 hi-D2}与某些地图u（x， t）、u（x， t）将与较少的运动密切相关，这是E. De Giorgi引入的一种新的运动。 ... More Tachikawa从谐波图（Eells-Sampson equation）到n尺寸尺度尺度的热型方程的构造弱解决方案（Eells-Sampson equation）到非紧凑型Riemannian流形（D 7 μ u（/）μ t μ u-u）|Du（Du）|イイD12イエD1 = 0）。莫弗，他提出了一个较弱的解决方案是能量功能的最小运动，与上一个问题有关， Nagasawa和Tachikawa研究了非紧凑完整的Riemannian Manifolds之间的谐波地图。特别是，他们认为谐波地图具有一定的非性别状况，并获得后续的非存在结果。“Let N be a Handamard manifold whose sectional curvManagement at a point p do not decay faster than Dist D1-2”（p， p， p，D20，D2） for some fixed point p。Then there is no entire Harmonic maps from Ry D1 m Ry D1 into N which satisfies a certain non-Degeneracy condition。“Nagasawa在Navier-Strokes equation on a Riemannian manifold using the above method上构建了一个较弱的解决方案。Moreover，他对较弱的解决方案进行了估计，该方案的结构高于并获得了新的部分规律性估计。他构造了一个关于高血压金茨堡兰道方程的弱解，也研究了他们的数字。Less（低）

The aim of this research is to study the solutions of the nonlinear partial differential equations which are closely related to variational problems by using calculus of variations and time-discretization schemes. More precisely, for some variational functional F(u) we treat the partial differential equation ィイD7∂u(/)∂tィエD7-(the Euler-Language equation of F) = 0. To construct weak solutions for the above equation we proceed as follows. We consider the functional GィイD2nィエD2(u) = ∫ィイD7|u-uィイD2nィエD2-1|(/)2hィエD7dx+F(u), and define uィイD2nィエD2 as a minimizer of GィイD2nィエD2(u) successively. Combining {uィイD2nィエD2} by line segments we construct a approximate solution uィイD2hィエD2(x, t). Finally, under some conditions, we prove that uィイD2hィエD2(x, t) converge to a weak solution. On the other hand, id the limit of the sequence of maps {uィイD2hィエD2}converges to some map u(x, t), u(x, t) will be closely related to minimizing movement which is a new notion introduced by E.De Giorgi.Using the above method … More , Tachikawa constructed weak solutions of the heat-type equations for harmonic maps (Eells-Sampson equation) from noncompact Riemannian manifolds into the n-dimensional spheres (ィイD7∂u(/)∂tィエD7-Δu-u|Du|ィイD12ィエD1 = 0). Moreover, he proved that the weak solutions are minimizing movements of the energy functionals.Related to the above problem, Nagasawa and Tachikawa studied harmonic maps between noncompact complete Riemannian manifolds. Especially, they considered harmonic maps with a certain non-degeneracy condition and get the following nonexistence result. "Let N be a Handamard manifold whose sectional curvatures at a point p do not decay faster than distィイD1-2ィエD1"(p, pィイD20ィエD2) for some fixed point pィイD20ィエD2. Then there is no entire harmonic maps from RィイD1mィエD1 into N which satisfies a certain non-degeneracy condition."Nagasawa constructed a weak solution of the Navier-Strokes equation on a Riemannian manifold using the above method. Moreover, he sharpened the energy estimates on the weak solutions constructed as above and got a new partial regularity estimates. He constructed weak solutions of the hyperbolic Ginzburg Landau equations too and studied them numerically. Less