New conservation laws of energy and C-D inequalities in continua with microstructure

2001 ◽  
Vol 22 (2) ◽  
pp. 135-143 ◽  
Author(s):  
Dai Tian-min
Keyword(s):  
Conservation Laws ◽  
Continua With Microstructure ◽  
Conservation Laws Of Energy

Long-time shallow-water equations with a varying bottom

1997 ◽  
Vol 349 ◽  
pp. 173-189 ◽  
Author(s):  
ROBERTO CAMASSA ◽  
DARRYL D. HOLM ◽  
C. DAVID LEVERMORE
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Euler Equations ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Three Dimensional ◽  
Local Conservation ◽  
First Order ◽  
Long Time ◽  
Conservation Laws Of Energy

We present and discuss new shallow-water equations that model the long-time effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid possessing a free surface and moving under the force of gravity. We consider the regime where the Froude number ε is much smaller than the aspect ratio δ of the shallow domain. The new equations are obtained from the ε→0 limit of the Euler equations (the rigid-lid approximation) at the first order of an asymptotic expansion in δ2. These equations possess local conservation laws of energy and vorticity which reflect exact layer mean conservation laws of the three-dimensional Euler equations. In addition, they convect potential vorticity and have a Hamilton's principle formulation. We contrast them with the Green–Naghdi equations.

Bosonic Symmetries, Solutions, and Conservation Laws for the Dirac Equation with Nonzero Mass

2013 ◽  
Vol 58 (6) ◽  
pp. 523-533 ◽  
Author(s):  
V.M. Simulik ◽  
◽  
I.Yu. Krivsky ◽  
I.L. Lamer ◽  
◽  
...  
Keyword(s):  
Dirac Equation ◽  
Conservation Laws
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