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.
In this article, we consider the random sampling in the image space
V$$ V $$ of an idempotent integral operator on mixed Lebesgue space
Lp,q()ℝn+1$$ {L}^{p,q}\left({\mathbb{R}}^{n+1}\right) $$. We assume some decay and regularity conditions on the integral kernel and show that the bounded functions in
V$$ V $$ can be approximated by an element in a finite‐dimensional subspace of
V$$ V $$ on
CR,S=[]−R2,R2n×[]−S2,S2$$ {C}_{R,S}={\left[-\frac{R}{2},\frac{R}{2}\right]}^n\times \left[-\frac{S}{2},\frac{S}{2}\right] $$. Consequently, we show that the set of bounded functions concentrated on
CR,S$$ {C}_{R,S} $$ is totally bounded and prove with an overwhelming probability that the random sample set uniformly distributed over
CR,S$$ {C}_{R,S} $$ is a stable set of sampling for the set of concentrated functions on
CR,S$$ {C}_{R,S} $$. Further, we propose an iterative scheme to reconstruct the concentrated functions from their random measurements.
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