2007
DOI: 10.1512/iumj.2007.56.2891
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Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $l_p$ initial data

Abstract: We prove the existence and Gevrey regularity of local solutions of the three dimensional periodic Navier-Stokes equations in case the sequence of Fourier coefficients of the initial data lies in an appropriate weighted p space. Our work is motivated by that of Foias and Temam ([10]) and we obtain a generalization of their result. In particular, our analysis allows for initial data that are less smooth than in [10] and can also have infinite energy. Our main tool is a variant of the Young convolution inequality… Show more

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Cited by 45 publications
(46 citation statements)
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“…Let us begin with analyticity results of this paper. Compared to previous works by [15,23,37] (derivative estimations), [4] (l p space on T 3 ), and [25,26] (complexified equations), we are able to establish analyticity of the Navier-Stokes equations by obtaining Gevrey estimates in Besov spacesḂ . This approach enables one to avoid cumbersome recursive estimation of higher order derivatives.…”
Section: Introduction and Statement Of Main Resultsmentioning
confidence: 67%
See 1 more Smart Citation
“…Let us begin with analyticity results of this paper. Compared to previous works by [15,23,37] (derivative estimations), [4] (l p space on T 3 ), and [25,26] (complexified equations), we are able to establish analyticity of the Navier-Stokes equations by obtaining Gevrey estimates in Besov spacesḂ . This approach enables one to avoid cumbersome recursive estimation of higher order derivatives.…”
Section: Introduction and Statement Of Main Resultsmentioning
confidence: 67%
“…It is well-known that regular solutions of many dissipative equations, such as the NavierStokes equations, the Kuramoto-Sivashinsky equation, the surface quasi-geostrophic equation and the Smoluchowski equation are in fact analytic, in both space and time variables [4,14,18,36,49]. In fluid-dynamics, the space analyticity radius has an important physical interpretation: at this length scale the viscous effects and the (nonlinear) inertial effects are roughly comparable.…”
Section: Introduction and Statement Of Main Resultsmentioning
confidence: 99%
“…As far as we know, the treatment of analyticity in dimensions 3 and higher in case of the KSE is new. In our analysis, we regard the KSE as an evolution equation in a suitable Banach space as in [11,13,18], and more recently in [1], and follow the variation of parameters approach. A noteworthy feature of this approach is the use of generalized Gevrey norms.…”
Section: The Kuramoto-sivashinsky Equation (Kse) Ismentioning
confidence: 99%
“…The method in this paper is similar to the one developed in [2] for the 3D space-periodic NSE and in [24] and [25] for the d-dimensional NSE defined on the whole space, where the requisite estimates on the nonlinear term are based on some simple convolution inequalities. We prove the existence of a Gevrey regular mild solution in Fourier space on a time interval [0, T ], which, by the Paley-Wiener theorem is a strong solution on any subinterval [t 0 , T ].…”
Section: The Kuramoto-sivashinsky Equation (Kse) Ismentioning
confidence: 99%
“…Related results in L p spaces were obtained in [27,28]. This approach has subsequently been revisited using more modern techniques in a variety of function spaces (see, e.g., [1,[3][4][5]21,30,33,34]). An alternative approach to the problem in L p spaces where p ∈ (3, ∞] was developed in [16,18,24] and is carried out entirely in physical space.…”
Section: Introductionmentioning
confidence: 99%