SHARP KATO SMOOTHING PROPERTIES OF WEAKLY DISSIPATED KDV EQUATIONS WITH VARIABLE COEFFICIENTS ON A PERIODIC DOMAIN.

It is well known that the solutions of the Cauchy problem of the Korteweg-de Vries (KdV) equation on a periodic domain T, u_t + uu_x + u_xxx = 0, u(x, 0) = φ(x), x ∈ T, t ∈ R, possess neither the sharp Kato smoothing property, φ ∈ H^s(T) =⇒ ∂^s+1_xu ∈ L^∞_x (T, L² (0, T)), nor the Kato smoothing property, φ ∈ H^s(T) =⇒ u ∈ L 2 (0, T; Hs+1(T)). This paper shows that the solutions of the Cauchy problem of following weakly dissipated KdV equation with variable coefficients posed on a periodic domain T, u_t + uu_x + a(x, t)u_xxx − (g(x, t)u_x)_x = 0, u(x, 0) = φ(x), where a and g are given real-valued smooth functions periodic in x satisfying a(x, t) 6 ≠ 0, x ∈ T, t ≥ 0 and ∫_T g(x, t)/|a(x, t)|dx > 0 ∀t ≥ 0, possess the sharp Kato smoothing property, φ ∈ H^s(T) =⇒ ∂^s+1_xu ∈ L^∞_x (T, L²(0, T)), ∀ s ≥ 0, and the nonlinear part of its solution u possesses the strong Kato smoothing property, φ ∈ H^s(T) =⇒ (u − v) ∈ C([0, T]; H^s+1(T)), ∀ s > ½, and the sharp double Kato smoothing property, φ ∈ H^s(T) =⇒ ∂^s+2_x (u − v) ∈ L^∞_x(T, L²(0, T)), ∀ s > ½, with v being the solution of the linear problem v_t + a(x, t)v_xxx − (g(x, t)v_x)_x = 0, v(x, 0) = φ(x), x ∈ T, t > 0. [ABSTRACT FROM AUTHOR]