Global Classical Solutions for Nonlinear Evolution Equations by T Li, Yun-Mei Chen

By T Li, Yun-Mei Chen

This article represents the implications initially acquired via S. Lainerman, D. Christodoulou, Y. Choquet-Bruhat, T. Nishida and A. Matsumara at the international lifestyles of classical options to the Cauchy challenge with small preliminary information for nonlinear evolution equations.

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60) (R«), as m -»+ 00. 1, the approximate solution determined and u ^it) can be uniquely 13 n o n l in e a r h e a t e q u a t io n s M ^(i)6//l(0,T;//i+l(IR«)). 53), we obtain {ujt)} is bounded in L“ (0, T;//S+l(|R«)), is bounded in ¿ 2(0 , T;//^(R«)). 67) M^(i)------^ u'(t) weakly in ¿2(0, T; //^(R«)). 6). 55) and Am e ¿2(0, r;//^([R«)). 54). 7). Thus, we get the existence of solutions. M. 56). 73). This finishes the proof. 2, we have, with eventual modificatibn on a set with zero measure on [0, T ], M€ C([0,n;//5+l(|R«)).

Rn)) and {D*v,} (11:1 = 2) a Cauchy sequence in 1-2(0, T; //^(R”)). Hence, by the completeness of these spaces, there exists a function u = u(t,x) such that, as i-* + oo, Vi ^ u strongly in L“ (0, T, iy5-«- 3,oo(|R«))^ V,dJ V,- u strongly in L“ (0, T; d J m ( U: I = HiR”)). 20) that for any finite T q with Ok T q ^ T , we have (1 + i)"/2 Vi-♦ (1 + i)"/2w strongly in L“ (0, Tq; W^^-"-3*“ (1R")). “ (1R")). 19) holds. This finishes the proof. 3. 2) By the definition of Xg g g, if £ < l , t h e n 40 GLOBAL SOLUTIONS FOR NONLINEAR EVOLUTION EQUATIONS IIVII Hence, for any vg ^ j 00 < 1, V v G £ 7'.

For brevity we only use the notation [0, T ] in what follows. By definition it is easy to see that, for any v e Xg g j , we have v e L “ (0 , T ; VF'J-n- 3. 9) (1 + f)«/2 V e L “ (0 , T ; VF5-'»- 3. 11) £>* ve ¿2(0,7;//S(|R«)) (|jtl = 2). 12) Moreover, if v e X5 g j , then, for any finite T q with 0 < T q 5 , 7 (v - P), V V, ^ e Xs, g, 7 , Xg g j is a nonempty complete metric space.

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