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Normal parabolic equation corresponding to 3D Navier-Stokes systemтезисы доклада
- Автор: Fursikov A.V.
- Сборник: International conference ``Partial Differential Equations and Applications'' in Honour of Mark Vishik on the occasion of his 90th birthday, Collection of abstracts
- Тезисы
- Год издания: 2012
- Место издания: IITP Moscow
- Первая страница: 16
- Аннотация: Energy estimate is very important tool to study 3D Navier-Stokes system. Absents of such bound in phase space H1 is very serious obstacle to prove nonlocal existence of smooth solutions. Semilinear parabolic equation is called equation of normal type if its nonlinear term B satisfies the condition: vector B(v) is collinear to vector v for each v. Since the property B(v) is orthogonal to v implies energy estimate, equation of normal type does not satisfies energy estimate "in the most degree". That is why we hope that investigation of normal parabolic equations should make more clear a number problems connected with existence of nonlocal smooth solutions to 3D Navier-Stokes equations. In the talk we will start from Helmholtz equations that is analog of 3D Navier-Stokes system in which the curl of fluid velocity is unknown function. We will derive normal parabolic equations (NPE) corresponding to Helmholtz equatins and will prove that there exists explicit formula for solution to NPE with periodic boundary conditions. This helped us to investigate more less completely the structure of dynamical flow corresponding to NPE. Its phase space V can be decomposed on the set of stability M_(a); a > 0 (solutions with initial condition belonging to M_(a) tends to zero with prescribed rate e^{-at} as time t tends to infinity), set of explosions M+ (solutions with initial condition belonging to M+ blows up during finite time), and intermediate set MI(a) = (V \(M_(a))\M+. The exact description of all these sets will be given.
- Добавил в систему: Фурсиков Андрей Владимирович