Navier–Stokes Equations and Weighted Convolution Inequalities in Groups

Biswas, Animikh; Swanson, David

doi:10.1080/03605300903488747

Cited by 20 publications

(41 citation statements)

References 18 publications

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“…However, to the best of our knowledge, Theorems 3.1 and 3.2 (as well as the spaces introduced in (8)) are new in the general setting described in (1). In the particular case of the Navier-Stokes equations, Theorem 3.1 generalizes those in [5] and [6] in the sense that it allows for a larger class of initial data (see discussion in Section 4.1). Also, in the special case of the Navier-Stokes or the surface quasi-geostrophic equations, we can take p = ∞ in Theorem 3.2.…”

Section: Remarkmentioning

confidence: 90%

“…Here, and henceforth, we will denote by C any generic constant which may depend only on the fixed parameters, like 1 p ∞ or the ones occurring in P1-P3 or (6). Also, for v ∈ V, here (and henceforth) we denote by |v| the R + -valued function on R n defined by |v|(ξ ) = |v(ξ )|.…”

Section: Notation and Settingmentioning

confidence: 98%

“…Analyticity of solutions for the 3D Navier-Stokes equations in the space V 2,∞ was proven first in [42]; see also [6].…”

Section: Remarkmentioning

confidence: 98%

“…Since its introduction, Gevrey class technique has become a standard tool for studying analytic properties of solutions for a wide class of dissipative equations (see [9,17,4,25]). This was extended to establish analyticity of solutions to the NSE in L p spaces [25], and subsequently to initial data in certain distributional spaces [6]. In [46], it was shown how Gevrey norm estimates can be used to derive sharp upper and lower bounds for the (time) decay of higher order derivatives of solutions to the NSE.…”

mentioning

confidence: 98%

“…The operatorĎ is a densely defined "dissipative" operator whileB(ǔ,v) is a densely defined bilinear operator. The bilinear operators we consider (see (6)) will include those having the form…”

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confidence: 99%

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Gevrey regularity for a class of dissipative equations with applications to decay

Biswas

2012

Journal of Differential Equations

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In this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via "caloric extension". We apply our result to the Navier-Stokes equations, the surface quasi-geostrophic equations, the Kuramoto-Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier-Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation.

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Section: Remarkmentioning

confidence: 90%

Section: Notation and Settingmentioning

confidence: 98%

“…Analyticity of solutions for the 3D Navier-Stokes equations in the space V 2,∞ was proven first in [42]; see also [6].…”

Section: Remarkmentioning

confidence: 98%

mentioning

confidence: 98%

mentioning

confidence: 99%

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Gevrey regularity for a class of dissipative equations with applications to decay

Biswas

2012

Journal of Differential Equations

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Convolution on distribution spaces characterized by regularization

Kleiner

Hilfer

2023

Mathematische Nachrichten

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Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.

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On extremal domains and codomains for convolution of distributions and fractional calculus

Kleiner

Hilfer

2022

Monatsh Math

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It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz potentials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.

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Navier–Stokes Equations and Weighted Convolution Inequalities in Groups

Cited by 20 publications

References 18 publications

Gevrey regularity for a class of dissipative equations with applications to decay

Gevrey regularity for a class of dissipative equations with applications to decay

Convolution on distribution spaces characterized by regularization

On extremal domains and codomains for convolution of distributions and fractional calculus

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