The Kawahara and modified Kawahara equations are fifth-order KdV type equations that have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for the Kawahara equation in H s (R) with s > −7/4 and the local well-posedness for the modified Kawahara equation in H s (R) with s ≥ −1/4. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the [k; Z] multiplier norm method of Tao [14] and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.

The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally well posed in H −7/4 , using the ideas of anF s -type space [8]. Then we show that the equation is globally well posed in H s for s ≥ −7/4, using the ideas of the "I-method" [7].

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In this paper, we study 1 − α 2 (0 < α 1) minimization methods for signal and image reconstruction with impulsive noise removal. The data fitting term is based on 1 fidelity between the reconstruction output and the observational data, and the regularization term is based on 1 − α 2 nonconvex minimization of the reconstruction output or its total variation. Theoretically, we show that under the generalized restricted isometry property that the underlying signal or image can be recovered exactly. Numerical algorithms are also developed to solve the resulting optimization problems. Experimental results have shown that the proposed models and algorithms can recover signal or images under impulsive noise degradation, and their performance is better than that of the existing methods.

Abstract. Let µ be a non-negative Radon measure on R d which only satisfies some growth condition. In this paper, the authors obtain the boundedness of Calderón-Zygmund operators in the Hardy space H 1 (µ).