Global Existence And Partial Regularity For Heat Flow Of H-Systems

In this paper, we are concerned with the global existence and partial regularity of weak solution to the heat flow of H-systems and the structure of singularities (bubbles).The H-system equations come from the Differential Geometry and have a long history. It is related to minimal surface problem, Plateau problem and constant mean curvature problem and so on. This problem and the other problems came from Differential Geometry, such as harmonic maps, are mainly considered in Variational Method. There are three problems related to Variational Method in Hilbert's famous 23 problems. This indicated the importance of Variational Method and made it developed faster in 20 century. The discovery of new principles in Variational Method also made faster developments in Partial Differential Equation related to Differential Geometry.The method we used in our paper is heat flow method. This method is introduced by Eells and Sampson [35] in 1964 when they studied harmonic maps. The nonlinear term which occurs in our case is of the same order as the nonlinear term which occurs in the case of harmonic heat flows. This suggests that we can get similar results for them. However, they are quite different. The harmonic map heat flow is the gradient flow of the Dirichlet energy, which naturally satisfies the energy inequality. This is the key estimate for global existence and partial regularity. Unfortunately, it is still open whether smooth solutions to heat flow of H-systems satisfy the energy inequality property. This is the main difficulty in our study.There are some results on the global existence and partial regularity of weak solution to the heat flow of H-systems. In two-dimensional case, in 1990, O. Rey [76] has proved the existence of global regular solution to the equations with the assumption ||H||∞||u₀||∞< 1.In 2002, Chen-Levine [24] have shown the existence of a unique, regular solution to the equations up until the first singular time for a general initial and boundary data u₀∈H¹(Ω), x∈H^?(αΩ). In addition, assuming that the solution satisfies the energy inequality, they can get the existence of a global regular solution with exception of at most finite many singularities and discuss the bubble phenomenon of the singularities. There are only few results on in higher-dimensional case to our knowledge.We extend the former results in the following respects:(1) In two-dimensional case, we get rid of the extra assumption on Dirichlet energy inequality in Chen-Levine [24] and prove some kind of Dirichlet energy estimate related to time for the solutions. Then using the classic boot-strap method, we prove that there is a unique weak solution to the heat flow of H-systems for any finite time. This weak solution is smooth except for finite singular points for any finite time.(2) In two-dimensional case, for any finite singular time, we prove that all bubbles can only occur in the interior of the domain. We also give an example of finite time blowup. This is to say that there is finite time blowup for the solution to the equations in some special case.(3) In two-dimensional case, we obtain the exact profiles for these singular points at finite time. We prove the energy identity at any finite singular time. That is to say the energy before the blowup equals to the energy of the weak solution and the bubbles after the blowup. In Chen-Levine [24], they have only got an inequality about that. (4) In higher-dimensional, when the boundary date x∈C^2+α(?)) under the assumption of ||H||∞||u₀||∞< 1, or under the assumption of ||H||∞||u₀||∞≤1 and the energy of u₀ is small enough, we can prove global existence of unique regular solution to the equations. Under the assumption of the energy inequality, we can also prove the global existence of weak solution to the equations. This solution is regular except for finite singular times. At any singular time, we find the concentration phenomenon (bubble) as in two-dimensional case and there are only finite bubbles at every singular point.This paper is organized as follows:In Chapterâ… , we introduce the background of H-systems and heat flow of H-systems. We also briefly introduce the history of the harmonic heat flows.In Chapterâ…¡, we collect some related definitions and properties.In Chapterâ…¢, we consider the global existence and partial regularity of the weak solution to the heat flow of H-systems in two-dimensional. The method we used in analysis of concentration of energy has been used in Lin-Wang [59]. We find the P.S. sequences of the heat flow of H-systems and use the P.S. sequences to blow up bubbles.In Chapterâ…£, we consider the global existence and partial regularity of the weak solution to the heat flow of H-systems in higher-dimensional.