Results In Sobolev Spaces Research Articles

An application of the weighted discrete Hardy inequality

In a note published in 1925, G. H. Hardy stated the inequality \begin{equation*} \sum_{n=1}^\infty \left(\frac{1}{n}\sum_{k=1}^n a_k \right)^p \leq \left(\frac{p}{p-1}\right)^p \sum_{n=1}^\infty a_n^p, \end{equation*} for any non-negative sequence $\{a_n\}_{n \geq 1}$, and $p>1$. This inequality is known in the literature as the classical discrete Hardy inequality. It has been widely studied and several applications and new versions have been shown. In this work, we use a characterization of a weighted version of this inequality to exhibit a sufficient condition for the existence of solutions of the differential equation ${\rm div}\,{\bf u}=f$ in weighted Sobolev spaces over a certain plane irregular domain. The solvability of this equation is fundamental for the analysis of the Stokes equations. The proof follows from a local-to-global argument based on a certain decomposition of functions which is also of interest for its applications to other inequalities or related results in Sobolev spaces, such as the Korn inequality.

Time-decay and Strichartz estimates for the BBM equation on modulation spaces: Existence of local and global solutions

In this paper we derive time-decay and Strichartz estimates for the generalized Benjamin-Bona-Mahony equation on the framework of modulation spaces Mp,qs. We use these results to analyze the existence of local and global solutions of the corresponding Cauchy problem with rough data in modulation spaces. As consequence of the existence theorems in modulation spaces, some results in Sobolev spaces are derived.

Well-posedness for a perturbation of the KdV equation

This is a continuation of Carvajal and Panthee (Q Appl Math 74:571–594, 2016). In the present paper we study a dissipative versions of the Korteweg de Vries equation with rough initial data. By working in Bourgain’s type spaces we prove local well posedness results in Sobolev spaces of negative order.

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

AbstractIn this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .

Local well-posedness and stability of solitary waves for the two-component Dullin–Gottwald–Holm system

In this paper, we study the Cauchy problem for the two-component Dullin–Gottwald–Holm system with the initial data (u0,ρ0)∈Bp,rs(R)×Bp,rs−1(R),1≤p,r≤+∞, and s>max{1+1p,32,2−1p}, which generalizes some previous local well-posed results in Sobolev spaces. Then we prove that all the smooth solitary wave solutions are orbitally stable under small disturbances.

Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials

We consider the AKNS (Ablowitz-Kaup-Newell-Segur) operator on the unitinterval with potentials belonging to Sobolev spaces in the framework ofinverse spectral theory. Precise sets of eigenvalues are given in orderthat they, together with the knowledge of the potentials onthe side $(a,1)$ and partial informations on the potential on $(a-\varepsilon,a)$ for some arbitrary small$\varepsilon>0$,determine thepotentials entirely on $(0,1)$. Naturally, the smaller is $a$ andthe more partial informations are known, the less is the number of the neededeigenvalues.

The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order

In this paper we study some dissipative versions of the Korteweg de Vries equation with rough initial data. By working in Bourgain's type spaces we prove local and global well posedness results in Sobolev spaces of negative order. In particular, we prove the global well posedness of the Korteweg de Vries Burgers equation u t + u x x x - u x x + uu x = 0 in H s (R) for s > - 34/ - ½4.