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

The aim of this research is to study The solutions of The nonlinear partial differential equations这是密切相关的variational problems by使用calculus of variations andtime-discretization schemes. More precisely，for some variational functional F(u) we treat the partial differential equation∂u(/)∂t D7-(theEuler-Language equation of F) = 0. To construct weak solutions for the above equation we proceed asfollows. We consider the functional G e D2n e D2(u) =∫D7|u-u e D2n e d21 |(/)2h e D7dx+F(u)，and define u - D2n D2 as a minimizer of G - D2n D2(u) successively. Combining {u - D2n - D2} by linesegments we construct a approximate solution u D2h D2(x, t). Finally, under some conditions，we prove that u D2h D2(x, t) converge to a weak solution. On the other hand，id the limit of the sequence of maps {u - D2h - D2}converges to some map u(x, t), u（x, t）t)将关闭相关的最小运动，这是一个新的介绍E.DeGiorgi.Using the above method…more,Tachikawa constructed weak solutions of the heat-type equations for harmonic maps （Eells-Sampson）from noncompact Riemannian manifolds into the n-dimensional spheres（D7∂u(/)∂t∂D7-Δu-u|Du|∂D12∂D1 = 0）. Moreover，he proved that the weak solutions are minimizing movements of the energy functionals.Related to the weak solutions are minimizing movements of the energy functionals.Related to theabove 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让一个Handamard manifold whose sectional curvatures at a point p do notdecay faster than dist - 1-2 - D1"(p, pp - D20 - D2) for some fixed point p - D20 - D2. Then there is no entire harmonic maps from R - D1m - D1into N which satisfies a certain non-degeneracy condition.“Nagasawa constructed a weak solution of ”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 partialHe constructed weak solutions of the hyperbolic Ginzburg Landau equations tooand studied them numerically. Less

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