Well-posedness and critical index set of the cauchy problem for the coupled kdv-kdv systems on t

Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems {u(t) _a(1)u(xxx )- c(11)uu(x )+ c(12)vv(x)+ d(11)u(x)v + d(12)uv(x), v(t) +a(2)v(xxx )= c(21)uu(x) + c(22)vv(x) +d(21)u(x)v + d(22)uv(x), (0.1) (u, v) | (t=0) = (u(0), v(0)) posed on the periodic domain T in the following four spaces H-1(s):= H-0(s)(T) xH(0)(s)(T),H-2(s):= H-0(s)(T) x H-s(T), H-3(s):= H-s(T) xH(0)(s)(T), H-4(s):= H-s(T) xH(s)(T). The coefficients are assumed to satisfy a1a2 =6 0 and P i,j Fix k is an element of{1,2, 3, 4}. Then for any coefficients a(1), a(2), (c(ij)) and (d(ij)), it is shown that there exists a critical index s(k)* is an element of(-infinity, + infinity] such that system (0.1) is analytically locally well-posed in H-k (s )if s > s (k)* but weakly analytically ill-posed if s < s(k)*. Viewing s(k)* as a function of the coefficients, its range Ck is defined to be the critical index set for the analytical well-posedness of (0.1) in H-k (s ). By investigating some properties of the irrationality exponents of the real numbers and by establishing some sharp bilinear estimates in non-divergence form, we manage to identify C-1= {-1/2, infinity} boolean OR (alpha: <= 1/2 <= alpha 1}, C-q = {-1/2, 1/4, infinity}boolean OR {alpha : 1/2 <=alpha <= 1} for q= 2, 3, 4. In particular, these sets contain an open interval (1/2, 1). This is in sharp contrast to the R case in which the critical index set C for the analytical well-posedness of (0.1) in the space H-s(R) x H-s(R) consists of exactly four numbers: C = {- 13/12 , -3/4 , 0, 3/4}.