Tags » Math.AP
The Euler equations for three-dimensional incompressible inviscid fluid flow are
where is the velocity field, and is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the formal calculations here; in particular, we will apply inverse operators such as or formally, assuming that these inverses are well defined on the functions they are applied to. 696 more words
The wave equation is usually expressed in the form
where is a function of both time and space , with being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain with some other manifold and replacing the Laplacian with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). 1,149 more words
As in the previous post, all computations here are at the formal level only.
Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. 2,057 more words