scholarly journals Asymptotic analysis of the lubrication problem with nonstandard boundary conditions for microrotation

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2233-2247 ◽  
Author(s):  
Igor Pazanin
Keyword(s):  
Boundary Condition ◽  
Film Thickness ◽  
Asymptotic Analysis ◽  
Boundary Conditions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Type System ◽  
Effective Model ◽  
Asymptotic Model ◽  
Incompressible Micropolar Fluid

The purpose of this paper is to propose a new effective model describing lubrication process with incompressible micropolar fluid. Instead of usual zero Dirichlet boundary condition for the microrotation, we consider more general (and physically justified) type of boundary condition at the fluid-solid interface, linking the velocity and microrotation through a so-called boundary viscosity. Starting from the linearized micropolar equations, we derive the second-order effective model by means of the asymptotic analysis with respect to the film thickness. The resulting equations, in the form of the Brinkman-type system, clearly show the influence of new boundary conditions on the effective flow. We also discuss the rigorous justification of the obtained asymptotic model.

scholarly journals A note on boundary conditions in Euclidean gravity

2021 ◽  
pp. 2140004
Author(s):  
Edward Witten
Keyword(s):  
Boundary Condition ◽  
General Relativity ◽  
Perturbation Theory ◽  
Quantum Gravity ◽  
Boundary Conditions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Conformal Geometry ◽  
Extrinsic Curvature ◽  
Euclidean Signature

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018

scholarly journals Asymptotic analysis for a Vlasov–Fokker–Planck/Navier–Stokes system in a bounded domain

2021 ◽  
pp. 1-83
Author(s):  
Young-Pil Choi ◽  
Jinwook Jung
Keyword(s):  
Boundary Condition ◽  
Asymptotic Analysis ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Stokes System ◽  
Dirichlet Boundary ◽  
Euler System ◽  
Navier Stokes ◽  
Fluid System ◽  
Fokker Planck

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov–Fokker–Planck equation coupled with the compressible isentropic Navier–Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier–Stokes system. Our main strategy relies on the relative entropy argument based on the weak–strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier–Stokes system.

scholarly journals HIGHER ORDER TRACE RELATIONS FOR SCHRÖDINGER OPERATORS

1995 ◽  
Vol 07 (06) ◽  
pp. 893-922 ◽  
Author(s):  
F. GESZTESY ◽  
H. HOLDEN ◽  
B. SIMON ◽  
Z. ZHAO
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Trace Formula ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Trace Formulae ◽  
Derivatives Of

We extend the trace formula recently proven for general one-dimensional Schrödinger operators which obtains the potential V(x) from a function ξ(x, λ) by deriving trace relations computing moments of ξ(λ, x) dλ in terms of polynomials in the derivatives of V at x. We describe the relation of those polynomials to KdV invariants. We also discuss trace formulae for analogs of ξ associated with boundary conditions other than the Dirichlet boundary condition underlying ξ.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

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