Application of the Realization of Homogeneous Sobolev Spaces to Navier-Stokes

Strichartz Type Estimates for Fractional Heat Equations

We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the HardyLittlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing L∞x (R ) by BMOx(R) and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with...

Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces

Regularity of the solutions of elliptic systems in polyhedral domains

The solution of the Dirichlet problem relative to an elliptic system in a polyhedron has a complex singular behaviour near edges and vertices. Here, we show that this solution has a global regularity in appropriate weighted Sobolev spaces. Some useful embeddings of these spaces into classical Sobolev spaces are also established. As applications, we consider the Lamé, Stokes and Navier-Stokes sy...

Asymptotic Integration of Navier - Stokes Equations with Potential Forces . Ii . an Explicit Poincaré - Dulac Normal Form

We study the incompressible Navier–Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper Ann. Inst. H. Poincaré Anal. Non Linéaire, 4(1):1–47, 1987, produces a Poincaré–Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normaliza...

Stability of Discrete Stokes Operators in Fractional Sobolev Spaces

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the timedependent Stokes equations with a source term in L(0, T ;L(Ω)) and prove uniform estimates on the time derivative and di...