Navier-Stokes Equations and Nonlinear Functional Analysis by Roger Temam

By Roger Temam

This moment variation, just like the first, makes an attempt to reach as easily as attainable at a few primary difficulties within the Navier-Stokes equations within the following components: lifestyles, strong point, and regularity of suggestions in area dimensions and 3; huge time habit of options and attractors; and numerical research of the Navier-Stokes equations. due to the fact that e-book of the 1st version of those lectures in 1983, there was large examine within the region of inertial manifolds for Navier-Stokes equations. those advancements are addressed in a brand new part committed totally to inertial manifolds.

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2) ^(C)+vAum(C) dC + PmB(um(C)9um(C)) = Pmf, um(0) = Pmuo, where f e C , and um maps C (or an open subset of C) into PmHc = CvVi+ • • • +Cw m . 2) possesses a unique solution u^ defined in C in some neighborhood of the origin. 45). ii) As in the real case, we now prove some a priori estimates on um. 1) with Aum(Ç) ( ( ^ J p . (£))) + v |Au m (0| 2 +)) Therefore ~ | k , ( s e w ) | | 2 + vcos 9 |Au m (se i9 )| 2 2 as < 7 - 3) = - R e e'6(B(um(seie), lUse")), A i U « " » , + Ree" (/,Au m (se«»)>.

3. Outlines of the proofs. 2 can be found in the original papers or in the books of O. A. Ladyzhenskaya [1], J. L. Lions [1], [RT]. We finish this section with some outlines of the proofs which we need for the sequel. i) We implement a Galerkin method, using as a basis of H the eigenfunctions wj9jeN, of the operator A (cf. 17)). 41) Um=L Zjm(t)Wj, J= l um : [0, T]»-> Wm = the space spanned by w^ . . 44) 4 P m is the orthogonal projector in H onto Wm. 45) ^ + vAUn + PmBum = Pmf. 4 Due to the definition of the Wy's, P m is also the orthogonal projector on V^, in V, V , D(A),....

EXISTENCE AND UNIQUENESS THEOREMS (MOSTLY CLASSICAL RESULTS) 23 the same is true on (0, T*(u0)), but u will be a strong solution on the whole interval [0, T] if u satisfies some further regularity property. For instance, if a weak solution u belongs to L 4 (0, T; V), then u is a strong solution: indeed u is a strong solution on some interval (0, T*), T*^kT, Let us assume that the largest possible value of T* is V < T. Then we must have i* II / Ml limsup||u(0|| = + 00 . 24), which is valid on (0, T'-e) all e > 0, implies d_ -(l+||u(r)||2)^c(l+||u(r)||2)3, dt for and by the technique of Gronwall's lemma, l-h||u(r)|| 2 ^(l4-||u 0 || 2 )exp (c£ X (l-^||u(5)|| 2 ) 2 ds), 0

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