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Eventual partition of conserved quantities in wave motion. (English) Zbl 0591.35036
Summary: Let u be a classical solution to the wave equation in an odd number n of space dimensions, with compact spatial support at each fixed time. R. J. Duffin [ibid. 32, 386-391 (1970; Zbl 0223.35055)] uses the Paley- Wiener theorem of Fourier analysis to show that, after a finite time, the (conserved) energy of u is partitioned into equal kinetic and potential parts. The wave equation actually has \((n+2)(n+3)/2\) independent conserved quantities, one for each of the standard generators of the conformal group of \((n+1)\)-dimensional Minkowski space. Of concern in this paper is the ”zeroth inversional quantity” \(I_ 0\), which is commonly used to improve decay estimates which are obtained using conservation of energy. We use Duffin’s method to partition \(I_ 0\) into seven terms, each of which, after a finite time, is explicitly given as a constant-coefficient quadratic function of the time, E. C. Zachmanoglou [ibid. 39, 296-297 (1972; Zbl 0233.35059)] has shown that under the above assumptions if \(n\geq 3\), the spatial \(L^ 2\) norm of u is eventually constant. A consequence of the analysis here is a bound on this constant in terms of the energy and the radius of the support of the Cauchy data of u at a fixed time.

35L05 Wave equation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35Q99 Partial differential equations of mathematical physics and other areas of application
Full Text: DOI
[1] Dassios, G, Equipartition of energy for Maxwell’s equations, Quart. appl. math., 37, 465-469, (1980) · Zbl 0425.35082
[2] Duffin, R.J, Equipartition of energy in wave motion, J. math. anal. appl., 32, 386-391, (1970) · Zbl 0223.35055
[3] Lax, P.D; Phillips, R.S, Scattering theory, (1967), Academic Press New York · Zbl 0214.12002
[4] Strauss, W, Nonlinear invariant wave equations, (), 197-249
[5] Zachmanoglou, E.C, Integral constants in wave motion, J. math. anal. appl., 39, 296-297, (1972) · Zbl 0202.37104
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