2022-05-23T16:00:26Zhttps://repository.dl.itc.u-tokyo.ac.jp/oai
oai:repository.dl.itc.u-tokyo.ac.jp:000055722021-03-01T19:40:34Z
バナッハ空間上の半線型放物型発展方程式に対する確定節点Determining nodes for semilinear parabolic evolution equations in Banach spacesKakizawa, Ryohei11601410University of Tokyo (東京大学)博士(数理科学)We are concerned with the determination of the asymptotic behaviour of strong solutions to the initial-boundary value problem for general semilinear parabolic equations by the asymptotic behaviour of these strong solutions on a finite set. More precisely, if the asymptotic behaviour of the strong solution is known on a suitable finite set which is called determining nodes, then the asymptotic behaviour of the strong solution itself is entirely determined. We prove the above property by the energy method. Moreover, we are concerned with the determination of the asymptotic behaviour of mild solutions to the abstract initial value problem for semilinear parabolic evolution equations in Lp by the asymptotic behaviour of these mild solutions on a finite set. More precisely, if the asymptotic behaviour of the mild solution is known on determining nodes, then the asymptotic behaviour of the mild solution itself is entirely determined. Not only the asymptotic equivalence but also rate of monomial or exponential convergence can be clarified. We prove the above properties by the theory of analytic semigroups on Banach spaces. As an important application of sectorial operators, we give the linearized operator (Stokes operator) associated with the initial-boundary value problem for the Navier-Stokes equations in a multiply-connected bounded domain with the Navier-Dirichlet boundary condition. Furthermore, we study the asymptotic properties of stationary solutions to this problem. As for the existence and uniqueness, this problem has uniquely a stationary solution in (W2p)n satisfying Lp estimates for any n < p < ∞. The first result is obtained from resolvent estimates for the Stokes operator in L(p,σ) and the Banach fixed point theorem. On the asymptotic stability, the stationary solutions are asymptotically stable in L(p,σ) if they are small in (W1p)n. The second result is proved by the theory of analytic semigroups on Banach spaces.thesis2012-03-222012-03-22application/pdfapplication/pdf甲第28381号https://repository.dl.itc.u-tokyo.ac.jp/record/5572/files/KakizawaR_24_3_PhD_a.pdfhttps://repository.dl.itc.u-tokyo.ac.jp/record/5572/files/KakizawaR_24_3_PhD_b.pdfeng