Inequalities in Mechanics and Physics
Produktnummer:
18506be7d846fd4eb487645bf8dca8e7bd
| Autor: | Duvant, G. Lions, J. L. |
|---|---|
| Themengebiete: | Finite Ungleichung approximation calculus continuum mechanics duality elasticity equation function mathematical physics |
| Veröffentlichungsdatum: | 15.11.2011 |
| EAN: | 9783642661679 |
| Sprache: | Englisch |
| Seitenzahl: | 400 |
| Produktart: | Kartoniert / Broschiert |
| Verlag: | Springer Berlin |
Produktinformationen "Inequalities in Mechanics and Physics"
1. We begin by giving a simple example of a partial differential inequality that occurs in an elementary physics problem. We consider a fluid with pressure u(x, t) at the point x at the instant t that 3 occupies a region Q oflR bounded by a membrane r of negligible thickness that, however, is semi-permeable, i. e., a membrane that permits the fluid to enter Q freely but that prevents all outflow of fluid. One can prove then (cf. the details in Chapter 1, Section 2.2.1) that au (aZu azu aZu) (1) in Q, t>o, -a - du = g du = -a z + -a z + -a z t Xl X X3 z l g a given function, with boundary conditions in the form of inequalities u(X,t»o => au(x,t)/an=O, XEr, (2) u(x,t)=o => au(x,t)/an?:O, XEr, to which is added the initial condition (3) u(x,O)=uo(x). We note that conditions (2) are non linear; they imply that, at each fixed instant t, there exist on r two regions r~ and n where u(x, t) =0 and au (x, t)/an = 0, respectively. These regions are not prescribed; thus we deal with a "free boundary" problem.
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