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On the weak solutions to a shallow water equation. (English) Zbl 1048.35092
The authors obtain the existence of global-in-time weak solutions to the Cauchy problem for a one-dimensional formally integrable shallow-water equation of Camassa-Holm type. First, for an approximate viscous problem they establish uniform in viscosity energy estimates for the solution and its gradient. This allows to derive a priori one-sided supernorm and space-time higher-norm estimates for first-order derivatives for viscous solutions, and thus, as the solution of the original problem is obtained as the limit of viscous approximation, these estimates are also valid for the solution of the original problem. The results are applied to the construction of large-time asymptotics.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B40 Asymptotic behavior of solutions to PDEs
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[1] Alber, Lett Math Phys 32 pp 137– (1994)
[2] Calogero, J Math Phys 37 pp 2863– (1996)
[3] Camassa, Phys Rev Lett 71 pp 1661– (1993)
[4] Camassa, Adv Appl Math 31 pp 1– (1994)
[5] Constantin, Comm Partial Differential Equations 23 pp 1449– (1998)
[6] Constantin, Acta Math 181 pp 229– (1998)
[7] Constantin, Comm Pure Appl Math 51 pp 475– (1998)
[8] DiPerna, Invent Math 98 pp 511– (1989)
[9] DiPerna, Comm Math Phys 108 pp 667– (1987)
[10] Weak convergence methods for nonlinear partial differential equations. CBMS Regional Conference Series in Mathematics, 74. American Mathematical Society, Providence, R.I., 1990.
[11] Fuchssteiner, Phys D 4 pp 47– (1981)
[12] Hunter, Arch Rational Mech Anal 129 pp 355– (1995)
[13] Joly, Trans Amer Math Soc 347 pp 3921– (1995)
[14] Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Vol. 1. Oxford Lecture Series in Mathematics and Its Applications, 3. Clarendon, Oxford University Press, New York, 1996.
[15] Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and Its Applications, 10. Clarendon, Oxford University Press, New York, 1998. · Zbl 0908.76004
[16] McKean, Asian J Math 2 pp 867– (1998) · Zbl 0959.35140
[17] Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, 136-212. Research Notes in Mathematics, 39. Pitman, Boston-London, 1979.
[18] Linear and nonlinear waves. Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1974.
[19] Theory of viscous conservation laws. Some recent topics in conservation laws. 141-193. and , editors. Studies in Advanced Mathematics, 15. American Mathematical Society/International Press, 1999.
[20] ; On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Preprint, 2000.
[21] Lectures on the calculus of variations and optimal control theory. W. B. Saunders, Philadelphia-London-Toronto, 1969. · Zbl 0177.37801
[22] Zhang, Asymptotic Anal 18 pp 307– (1998)
[23] Zhang, Acta Math Sin (Engl Ser) 15 pp 115– (1999)
[24] Zhang, Comm Partial Differential Equations
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