Search results for " PD"
showing 10 items of 651 documents
Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators
2018
This paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonallyinvariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $%L^{1}(\Omega \times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $%\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show that, under very natural assumptions, if an …
Localized forms of the LBB condition and a posteriori estimates for incompressible media problems
2018
The inf–sup (or LBB) condition plays a crucial role in analysis of viscous flow problems and other problems related to incompressible media. In this paper, we deduce localized forms of this condition that contain a collection of local constants associated with subdomains instead of one global constant for the whole domain. Localized forms of the LBB inequality imply estimates of the distance to the set of divergence free fields. We use them and deduce fully computable bounds of the distance between approximate and exact solutions of boundary value problems arising in the theory of viscous incompressible fluids. The estimates are valid for approximations, which satisfy the incompressibility …
Solution properties of the incompressible Euler system with rough path advection
2021
The present paper aims to establish the local well-posedness of Euler's fluid equations on geometric rough paths. In particular, we consider the Euler equations for the incompressible flow of an ideal fluid whose Lagrangian transport velocity possesses an additional rough-in-time, divergence-free vector field. In recent work, we have demonstrated that this system can be derived from Clebsch and Hamilton-Pontryagin variational principles that possess a perturbative geometric rough path Lie-advection constraint. In this paper, we prove the local well-posedness of the system in $L^2$-Sobolev spaces $H^m$ with integer regularity $m\ge \lfloor d/2\rfloor+2$ and establish a Beale-Kato-Majda (BKM)…
Navier-Stokes equations : unpredictability even without butterfiles?
2018
Mathematically, the motion of a fluid is described by the so-called Navier-Stokes equations. In the spirit of Newtonian mechanics, these equations should determine the future motion of the fluid out of its initial state. However, despite the significant effort made for more than a century, this determinism has not yet been mathematically proved nor disproved. This paper offers a general perspective on the Navier-Stokes equations, the fourth millennium problem.
Analytic solutions of the Navier-Stokes equations
2001
We consider the time dependent incompressible Navier-Stokes equations on an half plane. For analytic initial data, existence and uniqueness of the solution are proved using the Abstract Cauchy-Kovalevskaya Theorem in Banach spaces. The time interval of existence is proved to be independent of the viscosity.
Strictly convex corners scatter
2017
We prove the absence of non-scattering energies for potentials in the plane having a corner of angle smaller than $\pi$. This extends the earlier result of Bl{\aa}sten, P\"aiv\"arinta and Sylvester who considered rectangular corners. In three dimensions, we prove a similar result for any potential with a circular conic corner whose opening angle is outside a countable subset of $(0,\pi)$.
Regularity properties for quasiminimizers of a $(p,q)$-Dirichlet integral
2021
Using a variational approach we study interior regularity for quasiminimizers of a $(p,q)$-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincar\'{e} inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally H\"{o}lder continuous and they satisfy Harnack inequality, the strong maximum principle, and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for H\"older continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we cons…
Pointwise characterizations of Hardy-Sobolev functions
2006
We establish simple pointwise characterizations of functions in the Hardy-Sobolev spaces within the range n/(n+1)<p <=1. In addition, classical Hardy inequalities are extended to the case p <= 1.
Higher Order Sobolev-Type Spaces on the Real Line
2014
This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.
Pointwise Inequalities for Sobolev Functions on Outward Cuspidal Domains
2019
Abstract We show that the 1st-order Sobolev spaces $W^{1,p}(\Omega _\psi ),$$1&lt;p\leq \infty ,$ on cuspidal symmetric domains $\Omega _\psi $ can be characterized via pointwise inequalities. In particular, they coincide with the Hajłasz–Sobolev spaces $M^{1,p}(\Omega _\psi )$.