Download Attractors of Evolution Equations by A.V. Babin and M.I. Vishik (Eds.) PDF

By A.V. Babin and M.I. Vishik (Eds.)

ISBN-10: 0444890041

ISBN-13: 9780444890047

Difficulties, principles and notions from the speculation of finite-dimensional dynamical platforms have penetrated deeply into the idea of infinite-dimensional structures and partial differential equations. From the perspective of the speculation of the dynamical structures, many scientists have investigated the evolutionary equations of mathematical physics. Such equations contain the Navier-Stokes procedure, magneto-hydrodynamics equations, reaction-diffusion equations, and damped semilinear wave equations. a result of contemporary efforts of many mathematicians, it's been confirmed that the attractor of the Navier-Stokes method, which draws (in a suitable sensible area) as t - # all trajectories of the program, is a compact finite-dimensional (in the feel of Hausdorff) set. higher and decrease bounds (in phrases of the Reynolds quantity) for the measurement of the attractor have been came across. those effects for the Navier-Stokes procedure have encouraged investigations of attractors of different equations of mathematical physics.

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Example text

Let us introduce the following notations Vo=L(R), PO By IIuIIv we 1 to the norm in W' . P1 . H=L,(n). IIv is equivalent 1 due to zero Dirichlet boundary conditions * Vo and :V the spaces conjugate to V, and V1 Chapter 1 36 respectively. These spaces can be by means of the scalar product in H identified with subspaces of the space (C:(R))* of distributions on R. It is well-known that ( Lp(R) ) * = Lp,(R), l/p;+l/p,=l. By Sobolev's embedding theorem W1 (R)=Vlc L (R) when l/p = l/pl-l/n. O P where q satisfies (9).

3. Let F2c F,, the embedding being continuous. Let a semigroup ( S t ) be ( F1,F1)-bounded uniformly for t) 0, uniformly (F2,F2)-bounded for finite t and (Fl,F2)-bounded for t > 0. Then (St) is uniformly (F2,F2)-boundedfor t 0. Proof. Let BeB(F2). Then BEB(F,) and due to the uniform in t t ~ ( F1,F,) -boundedness of ( St), StBc B1eB(F1) W t r 0. By the (F1,F2)-boundedness of ( S t ) for t > 0 , B2=SBcB(F2). Therefore SIStBc SIBl=B2 wtzo or 1 1 STBc B, W t 2 1. By the uniform (F2 ,F2 )-boundedness of ( S t ) for finite t, StBcB3=5(F2) W t E [0,1].

On setting we deduce from ( 2 6 ) and ( 2 7 ) that = zll 1 uN1l2- 3 gll 2- 3u,,ll,". C2+ P P 0 p,l u IIP1 . NV1 A 1= A + c,, 41 Section 3 Using these estimates to bound from above the right-hand side of (25), we obtain the differential inequality a t iiuN ii25 -IIU ii2+ iigii2+ 2c2. This inequality yields the estimate from which we conclude in particular that the solution uN(t) of the equation (22) exists for all t, O = t < + m , . Integrating (25) in t from 0 up to T , T = T, we deduce using (26) and (29) that z T IluN(r)l12 + 2p 1IuN(t)ll P1dt + 2p v1 0s 0 0 z 5 p2(h + C)IIUN(t)l12 t 211gil-II~~(t)ll )dt f IluN(O)Il2 4 0 5 IlU 0)1l2+ C,T.

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