scholarly journals Asymptotics Around the Degeneration Spot of Heat Equation Solution with Strong Degeneration

2011 ◽  
Vol 11 (1) ◽  
pp. 9-19
Author(s):  
A.V. Glushko ◽  
◽  
A.D. Baev ◽  
D.S. Shumeeva ◽  
◽  
...  
Keyword(s):  
Heat Equation ◽  
Equation Solution ◽  
Strong Degeneration

scholarly journals The Heat Equation Solution Near the Interface Between Two Media

2017 ◽  
Vol 24 (3) ◽  
pp. 339-352 ◽  
Author(s):  
Natalia T. Levashova ◽  
Olga A. Nikolaeva
Keyword(s):  
Heat Equation ◽  
Equation Solution

Boundary Layers for the Navier–Stokes Equations. The Case of a Characteristic Boundary

2008 ◽  
Vol 15 (3) ◽  
pp. 517-530
Author(s):  
Makram Hamouda ◽  
Roger Temam
Keyword(s):  
Singular Perturbation ◽  
Heat Equation ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Navier Stokes Equations ◽  
Equation Solution ◽  
Characteristic Boundary ◽  
Thin Region ◽  
Perturbation Problems

Abstract We prove the existence of a strong corrector for the linearized incompressible Navier–Stokes solution on a domain with characteristic boundary. This case is different from the noncharacteristic case considered in [Hamouda and Temam, Some singular perturbation problems related to the Navier–Stokes equations: Springer Verlag, 2006] and somehow physically more relevant. More precisely, we show that the linearized Navier–Stokes solutions behave like the Euler solutions except in a thin region, close to the boundary, where a certain heat equation solution is added (the corrector). Here, the Navier–Stokes equations are considered in an infinite channel of but our results still hold for more general bounded domains.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

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