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)v(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) (uv)vertical bar(t=0) = (u(0), v(0)) posed on the periodic domain T in the following four spaces H-1(s) := H-0(s) (T) x H-0(s)(T); H-2(s) := H-0(s) (T) x H-s (T), H-3(s) := H-s (T) x H-0(s)(T); H-4(s) := H-s(T) x H-s (T): The coefficients are assumed to satisfy a(1)a(2) not equal 0 and (i,j)Sigma(c(ij)(2)+ d(ij)(2)) > 0. 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 C-k 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}.