On the solution of the equation $u_t+u^n u_x+H(x,t,u)=0$
Authors:
K. T. Joseph and P. L. Sachdev
Journal:
Quart. Appl. Math. 52 (1994), 519-527
MSC:
Primary 35L65; Secondary 35Q53
DOI:
https://doi.org/10.1090/qam/1292202
MathSciNet review:
MR1292202
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Abstract: We consider the equation ${u_t} + {u^n}{u_x} + H\left ( x, t, u \right ) = 0$ and derive a transformation relating it to ${u_t} + {u^n}{u_x} = 0$ . Special cases of the equation appearing in applications are discussed. Initial value problems and asymptotic behaviour of the solution are studied.
C. Bardos, A. Y. Leroux, and J. C. Nedelec, First-order quasilinear equation with boundary condition, Comm. Partial Differential Equations 4, 1017β1037 (1979)
D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech. 11, 11β33 (1979)
D. G. Crighton and J. F. Scott, Asymptotic solutions of model equations in nonlinear acoustics, Philos. Trans. Roy. Soc. London Ser. A 292, 101β134 (1979)
C. M. Dafermos, Regularity and large time behaviour of solutions of conservation law without convexity condition, Proc. Roy. Soc. Edinburgh Sect. A 99, 201β239 (1985)
F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, Self-similar spin-up spin-down in a cylinder of small ratio of height to diameter, J. Fluid Mech. 234, 473β486 (1992)
E. Hopf, The partial differential equation ${u_t} + u{u_x} = \mu {u_{xx}}$ , Comm. Pure Appl. Math. 3, 201β230 (1950)
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10, 537β566 (1957)
P. Lefloch, Explicit formula for scalar non-linear conservation laws with boundary condition, Math. Methods Appl. Sci. 10, 265β287 (1988)
J. D. Murray, Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations, Siam J. Appl. Math. 19, 135β160 (1970)
J. D. Murray, Lectures on nonlinear differential equation models in biology, Oxford Univ. Press, New York, 1977
J. J. C. Nimmo and D. G. Crighton, Geometrical and diffusive effects in non-linear acoustic propagation over long ranges, Philos. Trans. Roy. Soc. London Ser. A 320, 1β35 (1986)
P. L. Sachdev, Nonlinear diffusive waves, Cambridge Univ. Press, Cambridge, 1987
P. L. Sachdev, V. G. Tikekar, and K. R. C. Nair, Evolution and decay of spherical and cylindrical N waves, J. Fluid Mech. 172, 347β371 (1986)
P. L. Sachdev, K. T. Joseph, and K. R. C. Nair, Exact N-wave solutions for the non-planar Burgers equation, Proc. Roy. Soc. London (A), to appear (1994)
C. C. Shih, Attenuation characteristics of nonlinear pressure waves propagating in pipes, Finite Amplitude Wave Effects in Fluids (Bjorno, ed.), IPC Sci. Technol. Press, Guildford, 1974, pp. 81β87
E. H. Wedemeyer, The unsteady flow within a spinning cylinder, J. Fluid Mech. 20, 383β399 (1964)
C. Bardos, A. Y. Leroux, and J. C. Nedelec, First-order quasilinear equation with boundary condition, Comm. Partial Differential Equations 4, 1017β1037 (1979)
D. G. Crighton, Model equations of nonlinear acoustics, Ann. Rev. Fluid Mech. 11, 11β33 (1979)
D. G. Crighton and J. F. Scott, Asymptotic solutions of model equations in nonlinear acoustics, Philos. Trans. Roy. Soc. London Ser. A 292, 101β134 (1979)
C. M. Dafermos, Regularity and large time behaviour of solutions of conservation law without convexity condition, Proc. Roy. Soc. Edinburgh Sect. A 99, 201β239 (1985)
F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, Self-similar spin-up spin-down in a cylinder of small ratio of height to diameter, J. Fluid Mech. 234, 473β486 (1992)
E. Hopf, The partial differential equation ${u_t} + u{u_x} = \mu {u_{xx}}$ , Comm. Pure Appl. Math. 3, 201β230 (1950)
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10, 537β566 (1957)
P. Lefloch, Explicit formula for scalar non-linear conservation laws with boundary condition, Math. Methods Appl. Sci. 10, 265β287 (1988)
J. D. Murray, Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations, Siam J. Appl. Math. 19, 135β160 (1970)
J. D. Murray, Lectures on nonlinear differential equation models in biology, Oxford Univ. Press, New York, 1977
J. J. C. Nimmo and D. G. Crighton, Geometrical and diffusive effects in non-linear acoustic propagation over long ranges, Philos. Trans. Roy. Soc. London Ser. A 320, 1β35 (1986)
P. L. Sachdev, Nonlinear diffusive waves, Cambridge Univ. Press, Cambridge, 1987
P. L. Sachdev, V. G. Tikekar, and K. R. C. Nair, Evolution and decay of spherical and cylindrical N waves, J. Fluid Mech. 172, 347β371 (1986)
P. L. Sachdev, K. T. Joseph, and K. R. C. Nair, Exact N-wave solutions for the non-planar Burgers equation, Proc. Roy. Soc. London (A), to appear (1994)
C. C. Shih, Attenuation characteristics of nonlinear pressure waves propagating in pipes, Finite Amplitude Wave Effects in Fluids (Bjorno, ed.), IPC Sci. Technol. Press, Guildford, 1974, pp. 81β87
E. H. Wedemeyer, The unsteady flow within a spinning cylinder, J. Fluid Mech. 20, 383β399 (1964)
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© Copyright 1994
American Mathematical Society
