Limits to extensions of Burgers’ equation
Authors:
Steven Nerney, Edward J. Schmahl and Z. E. Musielak
Journal:
Quart. Appl. Math. 54 (1996), 385-393
MSC:
Primary 35Q53; Secondary 35Q99
DOI:
https://doi.org/10.1090/qam/1388023
MathSciNet review:
MR1388023
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Abstract: The vector Burgers’ equation is extended to include pressure gradients and gravity. It is shown that within the framework of the Cole-Hopf transformation there are no physical solutions to this problem. This result is important because it clearly demonstrates that any extension of Burgers’ equation to more interesting physical situations is strongly limited.
S. Nerney, E. J. Schmahl, and Z. E. Musielak, Analytic solutions of the vector Burgers’ equation, Quart. Appl. Math. 54, 63–71 (1996)
K. B. Wolf, L. Hlavaty, and S, Steinberg, Non-linear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families, J. Math. Anal. Appl. 114, 340–359 (1986)
J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225–236 (1951)
E. Hopf, The partial differential equation ${u_t} + u{u_x} = \mu {u_{xx}}$, Comm. Pure Appl. Math. 3, 201–230 (1950)
E. R. Benton, Solutions illustrating the decay of dissipation layers in Burgers’ nonlinear diffusion equation, Phys. Fluids 10, 2113–2119 (1967)
S. Nerney, E. J. Schmahl, and Z. E. Musielak, Analytic solutions of the vector Burgers’ equation, Quart. Appl. Math. 54, 63–71 (1996)
K. B. Wolf, L. Hlavaty, and S, Steinberg, Non-linear differential equations as invariants under group action on coset bundles: Burgers and Korteweg-de Vries equation families, J. Math. Anal. Appl. 114, 340–359 (1986)
J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225–236 (1951)
E. Hopf, The partial differential equation ${u_t} + u{u_x} = \mu {u_{xx}}$, Comm. Pure Appl. Math. 3, 201–230 (1950)
E. R. Benton, Solutions illustrating the decay of dissipation layers in Burgers’ nonlinear diffusion equation, Phys. Fluids 10, 2113–2119 (1967)
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Article copyright:
© Copyright 1996
American Mathematical Society
