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We show there exists a nontrivial $H^1_0$ solution to the steady Stokes equation on the 2D exterior domain in the hyperbolic plane. Hence we show there is no Stokes paradox in the hyperbolic setting. We also show the existence of a nontrivial solution to the steady Navier-Stokes equation in the same setting, whereas the analogous problem is open in the Euclidean case., Comment: 35 pages

The problem for the stationary Navier–Stokes equation in 3D under finite Dirichlet norm is open. In this paper we answer the analogous question on the 3D hyperbolic space. We also address other dimensions and more general manifolds.

We consider the generalization of the Navier-Stokes equations from $\mathbb R^n$ to the Riemannian manifolds. There are inequivalent formulations of the Navier-Stokes equations on manifolds due to the different possibilities for the Laplacian operator acting on vector fields on a Riemannian manifold. We present several distinct arguments that indicate that the form of the equations proposed by Ebin and Marsden in 1970 should be adopted as the correct generalization of the Navier-Stokes to the Riemannian manifolds., Comment: 16 pages; published version. See version 1 for the details of the non-relativistic limit, which were omitted in the published version

We develop the asymptotic behavior for the solutions to the stationary Navier-Stokes equation in the exterior domain of the 2D hyperbolic space. More precisely, given the finite Dirichlet norm of the velocity, we show the velocity decays to $0$ at infinity. We also address the decay rate for the vorticity and the behavior of the pressure.

The Leray–Hopf solutions to the Navier–Stokes equation are known to be unique on R 2 . In our previous work, we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so the uniqueness can be restored.

A gasket-type standard leak element with a microfluidic channel was developed and measured. The microfluidic channel with a diameter of several micrometers and a length of one hundred micrometers was fabricated by using femtosecond laser micromachining. The conductances (

This article is dedicated to the proof of \(C^\alpha \) regularization effects of Hamilton–Jacobi equations. The proof is based on the De Giorgi method. The regularization is independent on the regularity of the Hamilton.

We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Holder continuous for large time.