Sobolev Interpolation Inequalities with Weights

In this article, for 0 ≤m<∞ and the index vectors q=(q_1,q_2 ,q_3 ),r=(r_1,r_2,r_3) where 1≤q_i≤∞,1<r_i<∞ and 1≤i≤3, we study new results of Navier-Stokes equations with Coriolis force in the rotational framework in mixed-norm Sobolev-Lorentz spaces H ̇^(m,r,q) (R^3), which are more general than the classical Sobolev spaces. We prove the existence and uniqueness of solutions to the Navier-Stokes equations (NSE) under Coriolis force in the spaces L^∞([0, T]; H ̇^(m,r,q) ) by using topological arguments, the fixed point argument and interpolation inequalities. We have achieved new results compared to previous research in the Navier-Stokes problems.

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Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

Reviews in Mathematical Physics ◽

10.1142/s0129055x21500069 ◽

2020 ◽

pp. 2150006

Author(s):

Denis Bonheure ◽

Jean Dolbeault ◽

Maria J. Esteban ◽

Ari Laptev ◽

Michael Loss

Keyword(s):

Magnetic Field ◽

Dimensional Sphere ◽

Two Dimensional ◽

Dimensional Torus ◽

Euclidean Spaces ◽

Hardy Inequalities ◽

Interpolation Inequalities ◽

Nonlinear Interpolation ◽

Magnetic Potentials ◽

Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

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Extension Theorems on Weighted Sobolev Spaces and Some Applications

Canadian Journal of Mathematics ◽

10.4153/cjm-2006-021-0 ◽

2006 ◽

Vol 58 (3) ◽

pp. 492-528 ◽

Cited By ~ 5

Author(s):

Seng-Kee Chua

Keyword(s):

Sobolev Spaces ◽

Weighted Sobolev Spaces ◽

Poincaré Inequality ◽

Poincare Inequality ◽

Extension Theorems ◽

Doubling Weight ◽

Interpolation Inequalities

AbstractWe extend the extension theorems to weighted Sobolev spaces on (ε, δ) domains with doubling weight w that satisfies a Poincaré inequality and such that w–1/p is locally Lp′. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

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