Conserved quantity partition for Dirac’s equation
Author:
Thomas P. Branson
Journal:
Quart. Appl. Math. 42 (1984), 179-191
MSC:
Primary 35Q20; Secondary 35L45, 81D25
DOI:
https://doi.org/10.1090/qam/745098
MathSciNet review:
745098
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Abstract: Let $M$ be the $(n + 1)$-dimensional Minkowski space, $n \ge 3$. The energy of a solution $\psi$ to Dirac’s equation in $M$ is a sum of $n$ terms, the $j$ th term depending on $\psi$ and the space derivative $\partial \psi /\partial {x_j}$. We show that if the Cauchy datum for $\psi$ is compactly supported, then each of these terms is eventually constant. Specifically, if $\psi$ is initially supported in the closed ball of radius $b$ about the origin in space $\left ( {{R^n}} \right )$, then for times $\left | t \right | \ge b$, the $j$th term is equal to the energy of the $j$th Riesz transform ${( - \Delta )^{ - 1/2}}(\partial /\partial {x_j})\psi$, which also solves Dirac’s equation.
- Thomas P. Branson, Eventual partition of conserved quantities in wave motion, J. Math. Anal. Appl. 96 (1983), no. 1, 54–62. MR 717494, DOI https://doi.org/10.1016/0022-247X%2883%2990027-6
- Thomas P. Branson, Eventual partition of conserved quantities for Maxwell’s equations, Arch. Rational Mech. Anal. 86 (1984), no. 4, 383–394. MR 759770, DOI https://doi.org/10.1007/BF00280034
- David G. Costa and Walter A. Strauss, Energy splitting, Quart. Appl. Math. 39 (1981/82), no. 3, 351–361. MR 636240, DOI https://doi.org/10.1090/S0033-569X-1981-0636240-0
- George Dassios, Equipartition of energy for Maxwell’s equations, Quart. Appl. Math. 37 (1979/80), no. 4, 465–469. MR 564738, DOI https://doi.org/10.1090/S0033-569X-1980-0564738-9
- R. J. Duffin, Equipartition of energy in wave motion, J. Math. Anal. Appl. 32 (1970), 386–391. MR 269190, DOI https://doi.org/10.1016/0022-247X%2870%2990304-5
R. Glassey, unpublished communication (1982)
- Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
- Walter A. Strauss, Nonlinear invariant wave equations, Invariant wave equations (Proc. “Ettore Majorana” Internat. School of Math. Phys., Erice, 1977) Lecture Notes in Phys., vol. 73, Springer, Berlin-New York, 1978, pp. 197–249. MR 498955
- E. C. Zachmanoglou, Integral constants in wave motion, J. Math. Anal. Appl. 39 (1972), 296–297. MR 308604, DOI https://doi.org/10.1016/0022-247X%2872%2990201-6
T. Branson, Eventual partition of conserved quantities in wave motion, J. Math. Anal. Appl., 96, 54–62 (1983)
T. Branson, Eventual partition of conserved quantities for Maxwell’s equations, Arch. Rational Mech. Anal., to appear
D. Costa and W. Strauss, Energy splitting, Quart. Appl. Math. 38, 351–361 (1981)
G. Dassios, Equipartition of energy for Maxwell’s equations, Quart. Appl. Math. 37, 465–469 (1980)
R. Duffin, Equipartition of energy in wave motion, J. Math. Anal. Appl. 32, 386–391 (1970)
R. Glassey, unpublished communication (1982)
P. Lax and R. Phillips, Scattering theory, Academic Press, New York, 1967
W. Strauss, Nonlinear invariant wave equations, in Lecture Notes in Physics, vol. 73, 197–249, Springer-Verlag, Berlin, 1978
E. Zachmanoglou, Integral constants in wave motion, J. Math. Anal. Appl. 39, 296–297 (1972)
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Article copyright:
© Copyright 1984
American Mathematical Society