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 enabling us to estimate the nonlinear term in weighted p norm.

Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

“…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.…”

“…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.…”

Existence and generalized Gevrey regularity of solutions to the Kuramoto–Sivashinsky equation in Rn

“…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.…”

“…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 ].…”

Local analyticity radii of solutions to the 3D Navier–Stokes equations with locally analytic forcing

“…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.…”