Wellposedness of the Cauchy problems for hyperbolic systems with large data
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants|
|Research Institution||Shizuoka University(2006)|
TANAKA Naoki Shizuoka University, Faculty of Science, Professor, 理学部, 教授 (00207119)
TAMURA Hideo Okayama University, Graduate school of Natural Science and Technology, Professor, 大学院自然科学研究科, 教授 (30022734)
ASAKURA Fumioki Osaka Electro-Communication University, Faculty of Engineering, Professor, 工学部, 教授 (20140238)
MATSUMOTO Toshitaka Hiroshima University, Graduate School of Science, Assistant, 大学院理学研究科, 助手 (20229561)
SOBUKAWA Takuya Okayama University, Faculty of Education, Associate Professor, 教育学部, 助教授 (60252946)
KATSUDA Atsushi Okayama University, Graduate School of Natural Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (60183779)
佐藤 亮太郎 岡山大学, 大学院・自然科学研究科, 教授 (50077913)
|Project Period (FY)
2004 – 2006
Completed(Fiscal Year 2006)
|Budget Amount *help
¥3,600,000 (Direct Cost : ¥3,600,000)
Fiscal Year 2006 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 2005 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 2004 : ¥1,300,000 (Direct Cost : ¥1,300,000)
|Keywords||semigroup of Lipschitz operators / evolution equation / wellposedness in the sense of Hadamard / subtangential condition / dissipativity condition / stability condition / approximation theorem for semigroups of operators / semigroup of operators / semilinear equation / quasilinear equation / Hadamard wellposedness / perturbation / analytic semigroup / fractional power / metric-like functional / semigroup of Lipschitz operators / Chernoff's product formula / quasilinear evolution equation / semilinear evolution equation / acoustic boundary condition / Wentzell boundary condition / non-densely defined operator / well-posedness / semigroup of lipschitz operators / comparison function / infinitesimal generator / abstract Cauchy problem / subtangential condition / dissipativity condition / integrated semigroup|
1. Semigroups of locally Lipschitz operators are characterized by subtangential conditions and semilinear stability conditions in terms of a family of metric-like functionals. The result is applied to the mixed problem for the complex Ginzburg-Landau equation.
2. It is interesting to compute solutions of Cauchy problems for partial differential equations numerically and to discuss the question of convergence which arises in that case. Such problems are treated by the theory of semigroups of operators. A convergence theorem and an approximation theorem for semigroups of Lipschitz operators are given. These theorems are applied to a semi-discrete approximation problem for a quasi-linear wave equation with damping and a finite difference method for a quasi-linear equation of Kirchhoff type.
3. An approximation theorem is given for abstract quasi-linear evolution equations in the sense of Hadamard. The result is applied to an approximation problem for a degenerate Kirchhoff equation.
4. The situation that the domains of differential operators are not dense in the underlying space arises when the mixed problems for certain partial differential equations in the space of continuous functions are studied. This leads to the study of the abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators. The result is applied to obtain the global wellposedness for quasi-linear wave equations of Kirchhoff type with acoustic boundary conditions and the local solvability of quasi-linear wave equations with Wentzell boundary conditions in the space of continuous functions.
Research Products (20results)