New approach to the Hadamard variational formula for the Green function of the Stokes equations

2014 ◽  
Vol 146 (1-2) ◽  
pp. 85-106 ◽  
Author(s):  
Erika Ushikoshi
Keyword(s):  
Green Function ◽  
Stokes Equations ◽  
Variational Formula ◽  
New Approach ◽  
The Green Function ◽  
Hadamard Variational Formula

scholarly journals Hadamard’s variational formula in terms of stress and strain tensors

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Björn Gustafsson ◽  
Ahmed Sebbar
Keyword(s):  
Green Function ◽  
Stress Tensor ◽  
Strain Tensor ◽  
Variational Formula ◽  
Scalar Fields ◽  
Action Functional ◽  
Lagrangian Action ◽  
The Green Function ◽  
Strain Tensors ◽  
Hadamard Variational Formula

AbstractStarting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

Wave propagation for the compressible Navier–Stokes equations

2015 ◽  
Vol 12 (02) ◽  
pp. 385-445 ◽  
Author(s):  
Tai-Ping Liu ◽  
Se Eun Noh
Keyword(s):  
Green Function ◽  
Stokes Equations ◽  
Low Frequency ◽  
Explicit Construction ◽  
Large Time Behavior ◽  
Energy Estimates ◽  
Navier Stokes ◽  
Time Behavior ◽  
Navier Stokes Equations ◽  
The Green Function

We establish the pointwise description of solutions to the isentropic Navier–Stokes equations for compressible fluids in three spatial dimensions. First, we give an explicit construction of the Green function for the linearized system. The Green function consists of singular waves, which dominate the short-time behavior, while the low frequency waves, the dissipative Huygens, diffusion and Riesz waves representing the large-time behavior. The nonlinear terms are treated by a suitable combination of energy estimates and pointwise estimates using the Duhamel's principle for the Green function.

The matrix of the Green function and the Green representation formula of the Stokes equations for a half space

2020 ◽  
Vol 31 (12) ◽  
pp. 2050099
Author(s):  
Teppei Kobayashi
Keyword(s):  
Green Function ◽  
Half Space ◽  
Stokes Equations ◽  
Representation Formula ◽  
Regular Form ◽  
The Matrix ◽  
The Green Function

In this paper, we consider the matrix of the Green function of the Stokes equations for a half space. To the best of author’s knowledge, the matrix of the Green function has been obtained for a half space in [Formula: see text]. But the matrix of the Green function for a half space in [Formula: see text] has not been obtained. Moreover, the author considers that the form of the matrix of the Green function in [Formula: see text] is irregular. In this paper, we obtain the regular form of the matrix of the Green function of the Stokes equations for a half space in [Formula: see text]. Moreover we obtain the Green representation formula of the Stokes equations for a half space in [Formula: see text].

Hadamard variational formula for eigenvalues of the Stokes equations and its application

2016 ◽  
Vol 368 (1-2) ◽  
pp. 877-884
Author(s):  
Shuichi Jimbo ◽  
Hideo Kozono ◽  
Yoshiaki Teramoto ◽  
Erika Ushikoshi
Keyword(s):  
Stokes Equations ◽  
Variational Formula ◽  
Hadamard Variational Formula

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

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