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In Part II we shall consider some problems of this kind. Now we are interested in estimating from above the numbers W n for the operators U t, vo. We consider two classes of problem 4. For the second class, these operators are merely continuous and bounded for example, in the case of problems of hyperbolic type. In either case, the numbers W n may be estimated as follows. The space An H consists of the elements U1 A A un, VI A We denote the norm of Ul A A Un by lIul A A unll.

It is known also see [9], [10], etc. Let us take n linearly independent elements Ui O E H and let Uj t be the corresponding solutions of 4. Formula 4. Although the system 4. Integrating 4.

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Usually when the existence problem for 4. Here we shall follow the same line. These inequalities make it possible to estimate the right hand side of 4. E Hand 4. The terms with Sk ; 0 appear in some applications. As for problems of hyperbolic type and many others the operators Vi and U t, vo are merely continuous and bounded.

We now summarize the results: Theorem 4. Then the corresponding estimates 4. From Theorems 4. Suppose that L t, vo , Vo E A, satisfies the inequalities 4. Then dimH A O. Thus we obtain! In order to get dimJ A 0 and s o. If L t satisfies the condition 4. N'" Sle 4. Usually in some problems of mathematical physics e. In this case we may put h,,. Semigroups generated by evolution equations 5 Introduction to Part II In Part II we consider abstract semi-linear evolution equations mainly of hyperbolic type. They generate semigroups of class AK. Evolution equations of parabolic type generate semigroups of class K.

We devote to them only the short Chapter 6, as semi-linear parabolic equations are expounded in a comparatively complete literature. Many publications are devoted to the Navier-Stokes equations which generate in the two-dimensional case semigroups of class K. Here we give some comparatively recent results [11] concerning majorants for the number N 1 and for the fractal dimension of invariant bounded sets which are better for small viscosity v than before. On the other hand, quasi-linear parabolic equations of general form also generate semigroups of class K, but the presentation of this material requires a separate publication.

For this purpose we need to use results on the global unique solvability of boundary value problems for these equations and estimates of a local type. They may be found in the monograph [15]; more recent results are described in the survey [16], which also contains a list of publications. It is, nevertheless, necessary to put this subject in the framework of the theory of semigroups choosing the phase spaces correctly. Let H be a complete separable Hilbert space, ',.

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Moreover, A is self-adjoint, positive definite and its inverse A-I is completely continuous. Clearly, Ho A H.

Attractors for Semi-groups and Evolution Equations (Lezioni Lincee)

But the dependence on A will be explicitly indicated in Chapter 7, where we deal with operators depending on a parameter. In the case of homogeneous sticking boundary conditions i. So the following Theorem holds: Theorem 6. In the case m 3 with either sticking or periodic conditions the majorants of number N I for invariant sets A bounded in HI have the form 6.

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But for the Navier-Stokes equations and for some other partial differential equations it is possible to evaluate d ' A for any s using theorems from Chapter 4. In addition, they also depend on O. Below we shall formulate conditions for f under which the problem 7. By using the procedure of Chapter 5, for an operator A defined in a dense domain, we introduce a space-scale H 6 A , s E R. To apply the results of Part I, we shall reformulate the problem 7.

In this variant the problem 7. The vector function v t, a is connected with the solution v t of 7. It is easy to prove that, if v t is a solution of 7. In terms of the components Vo t, a , VI t, a of 17 t, a , the system 7. The scalar product in H",a will be denoted by the symbol. Under the conditions 7. As the phase space for problem 7. From 7. The latter enjoys all the properties required by this procedure.

Moreover ao a u,V x. We begin our analysis of the problem 7.

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We want to prove the solvability of 7. The main energy relation for the problem is 2 2. By using 7. In coordmates. The following relation holds: 1d 2 -2 -dt lI u t, a lIx. It is the result of the multiplication of 7. In fact, for an arbitrary u E X, and t1 a: C a: t7, where C a: is defined in 7. The essential things are: a the requirements 7. Now let us go on to prove the unique solvability of the problem 7. We shall call a weak solution of the problem 7.

It is easy to prove the following theorem Theorem 7.

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We shall apply this theorem to linear and nonlinear problems as a means of identifying their solutions when we have preliminary and 50 Semigroups generated by evolution equations incomplete information, but when we know that they are weak solutions of a problem of the type 7. Let us prove the following existence theorem: Theorem 7. For almost all t it satisfies the equation 7.

For this solution the estimates 7. Any weak solution of problem 7. It is easy to prove that all the statements of Theorem 7. As in the finite-dimensional case, the solution u t of problem 7. But, otherwise, the representation 7. Problem 7. The first component uo t of the solution u t is the solution u t of problem 7.

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Therefore from Theorem 7.