2003 Fiscal Year Final Research Report Summary
Well-posedness and approximation of Cauchy problems for hyperbolic systems
| Project/Area Number |
14540175
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | OKAYAMA UNIVERSITY |
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Principal Investigator |
TANAKA Naoki Okayama University, Associate Professor, 大学院・自然科学研究科, 助教授 (00207119)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Toshitaka Hiroshima University, Assistant, 大学院・理学研究科, 助手 (20229561)
KOBAYASHI Yoshikazu Niigata University, Professor, 工学部, 教授 (80092691)
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| Project Period (FY) |
2002 – 2003
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| Keywords | abstract quasi-linear equation / Hadamard well-posedness / stability / conservation law / regularized semigroup / generator / Lipschitz semigroup / integrated semigroup |
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| Research Abstract |
1. Cauchy problems for hyperbolic systems of conservation laws : A class of weak solutions is introduced for the systems of nonlinear conservation laws for which the shock and rarefaction curves coincide and an analogue of the classical theorem Kruzhkov is given for such systems.
2. Semigroups of locally Lipschitz operators : The continuous infinitesimal generators of semigroups of locally Lipschitz operators are characterized by the dissipative conditions defined by means of metric-like functionals and the so-called subtangential conditions.
3. Integrated semigroups : (1) A class of perturbing operators is introduced for locally Lipschitz continuous integrated semigroups, and some perturbation theorems are given for such integrated semigroups. (2) Nonlinear perturbations of integrated semigroups are treated from the viewpoint of nonlinear semigroup theory and a characterization of nonlinear semigroups is discussed.
4. Evolution operators generated by non-densely defined operators : It is
… More
shown that an evolution operator is generated by a family of closed linear operators whose common domain is not necessarily dense in the underlying Banach space, under the stability condition from the viewpoint of finite approximations.
5. Abstract Cauchy problems for quasi-linear evolution equations in the sense of Hadamard : The notion of well-posedness in the sense of Hadamard is introduced for abstract quasi-linear evolution equations of degenerate type. A new type of stability condition is also introduced from the viewpoint of finite difference approximations. The obtained theorem is applied to the local solvability of a degenerate Kirchhoff equation with nonlinear perturbation.
6. Abstract quasilinear equations of second order with Wentzell boundary conditions : A general framework is introduced to treat abstract quasilinear equations of-second order with Wentzell boundary conditions. The result is applied to a wave equation for a second order quasilinear differential operator in the continuous function space with Wentzell boundary condition. Less
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