GLOBAL ATTRACTOR FOR WEAKLY DAMPED, FORCED mKdV EQUATION BELOW ENERGY SPACE

Abstract

We prove the existence of the global attractor inḢ s , s > 11/12 for the weakly damped and forced mKdV on the one dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space is established. We directly apply the I-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.

Global attractor for the periodic generalized Korteweg-De Vries equation through smoothing

“…Later, Erdogan and Tzirakis showed in [11] using smoothing that (2) has a global attractor in L2 that is compact in H s , s ∈ (0, 1), with forcing in L2 . Moving on to the mKdV, Goyal [16] proved an analogous statement to those of Tsugawa and Yang by proving that the forced and damped mKdV possesses a global attractor in Ḣs for 1 > s > 11/12 with forcing f ∈ L2 ∩ H 1 , later improved to 1 > s > 9/10 in [17].…”

“…Proof. This proof is essentially the same as in [16,23], but for the sake of completeness we will include it. We assume that n ≥ 4 (for otherwise the bounds are strictly better),…”

“…In particular, we adopt their analysis of the resonant set described in the beginning of Section 3, as well as Lemmas 3.2 and 3.3-although we have changed the statement to Lemma 3.2 and the proofs to both. We overcome obstacles involved in the application of the guage transformation to the forcing term, as employed in [16], at the cost of a much longer local existence proof. We then apply a normal form transformation and perform an analysis on the resulting terms.…”

“…for z ∈ X 0,1/3 . Applying Hölders in exactly the same way as in the X s0,−1/2 case, and noting that the z bound is in X 0,1/3 , we arrive at (16) with a small power of T .…”

“…Local well-posedness of (8). In this section we apply the idea used in [16] to prove well-posedness. We begin by proving that the non-linear Duhamel term is in Y 1 .…”