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Global existence and blow-up for a shallow water equation. (English) Zbl 0918.35005

An interesting phenomenon in water channels is the appearance of waves with length much greater than the depth of the water. Recently, R. Camassa and D. Holm proposed a new model for the same phenomenon: \[ \begin{cases} u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx},\qquad t>0,\quad & x\in\mathbb{R},\\ u(0, x)= u_0(x),\quad & x\in\mathbb{R}.\end{cases}\tag{1} \] The variable \(u(t,x)\) in (1) represents the fluid velocity at time \(t\) in the \(x\) direction in appropriate nondimensional units (or, equivalently, the height of the water’s free surface above a flat bottom).
The aim of this paper is to prove local well-posedness of strong solutions to (1) for a large class of initial data, and to analyze global existence and blow-up phenomena. In addition, we introduce the notion of weak solutions to (1) suitable for soliton interaction.

MSC:

35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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References:

[1] M.S. Alber - R. Camassa - D. Holm - J.E. Marsden , On the link between umbilic geodesics and soliton solutions of nonlinear PDE’s , Proc. Roy. Soc. London , Ser. A 450 ( 1995 ), 677 - 692 . MR 1356178 | Zbl 0835.35125 · Zbl 0835.35125
[2] M.S. Alber - R. Camassa - D. Holm - J.E. Marsden , The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s , Lett. Math. Phys. 32 ( 1994 ), 137 - 151 . MR 1296383 | Zbl 0808.35124 · Zbl 0808.35124
[3] B. Benjamin - J.L. Bona - J.J. Mahony , Model equations for long waves in nonlinear dispersive systems , Philos. Trans. Roy. Soc. London , Ser. A 272 ( 1972 ), 47 - 78 . MR 427868 | Zbl 0229.35013 · Zbl 0229.35013
[4] J.L. Bona - W.G. Pritchard - L.R. Scott , Solitary wave interaction , Phys. Fluids 23 ( 1980 ), 438 - 441 . Zbl 0425.76019 · Zbl 0425.76019
[5] J. Bourgain , Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II , Geom. Funct. Anal. 3 ( 1993 ), 209 - 262 . MR 1215780 | Zbl 0787.35098 · Zbl 0787.35098
[6] F. Calogero , An integrable Hamiltonian system , Phys. Lett. A 201 ( 1995 ), 306 - 310 . MR 1331829 | Zbl 1020.37524 · Zbl 1020.37524
[7] F. Calogero - J.P. Francoise , A completely integrable Hamiltonian system , J. Math. Phys. 37 ( 1996 ), 2863 - 2871 . MR 1390240 | Zbl 0864.58025 · Zbl 0864.58025
[8] F. Calogero - J.F. Vandiejen , Solvable quantum version of an integrable Hamiltonian system , J. Math. Phys. 37 ( 1996 ), 4243 - 4251 . MR 1408090 | Zbl 0863.58022 · Zbl 0863.58022
[9] R. Camassa - D. Holm , An integrable shallow water equation with peaked solitons , Phys. Rev. Lett. 71 ( 1993 ), 1661 - 1664 . MR 1234453 | Zbl 0972.35521 · Zbl 0972.35521
[10] R. Camassa - D. Holm - J. Hyman , A new integrable shallow water equation , Adv. Appl. Mech. 31 ( 1994 ), 1 - 33 . Zbl 0808.76011 · Zbl 0808.76011
[11] A. Constantin , The Hamiltonian structure of the Camassa-Holm equation , Expositiones Math. 15 ( 1997 ), 53 - 85 . MR 1438436 | Zbl 0881.35094 · Zbl 0881.35094
[12] A. Constantin , On the Cauchy problem for the periodic Camassa-Holm equation , J. Differential Equations 141 ( 1997 ), 218 - 235 . MR 1488351 | Zbl 0889.35022 · Zbl 0889.35022
[13] A. Constantin - J. Escher , Well-posedness and existence of global solutions for a periodic quasi-linear hyperbolic equation , Comm. Pure Appl. Math. 51 ( 1998 ), 475 - 504 . MR 1604278 | Zbl 0934.35153 · Zbl 0934.35153
[14] F. Cooper - H. Shepard , Solitons in the Camassa-Holm shallow water equation , Phys. Lett. A 194 ( 1994 ), 246 - 250 . MR 1301972 | Zbl 0961.76512 · Zbl 0961.76512
[15] R.K. Dodd - J.C. Eilbeck - J.D. Gibbon - H.C. Morris , Solitons and Nonlinear Wave Equations , Academic Press , New York , 1984 . MR 696935 | Zbl 0496.35001 · Zbl 0496.35001
[16] A. Fokas - B. Fuchssteiner , Symplectic structures, their Bäcklund transformation and hereditary symmetries , Phys. D 4 ( 1981 ), 47 - 66 . MR 636470 · Zbl 1194.37114
[17] B. Fuchssteiner , Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation , Phys. D 95 ( 1996 ), 296 - 343 . MR 1406283 | Zbl 0900.35345 · Zbl 0900.35345
[18] D. Gilbarg - N.S. Trudinger , Elliptic Partial Differential Equations of Second Order , Springer Verlag , Berlin , 1977 . MR 473443 | Zbl 0361.35003 · Zbl 0361.35003
[19] T. Kato , Quasi-linear equations of evolution, with applications to partial differential equations , In: ” Spectral Theory and Differential Equations ”, 448 , Springer Lecture Notes in Mathematics , 1975 , pp. 25 - 70 . MR 407477 | Zbl 0315.35077 · Zbl 0315.35077
[20] T. Kato , On the Cauchy problem for the (generalized) Korteweg-de Vries equation , Stud. Appl. Math. 8 ( 1983 ), 93 - 126 . MR 759907 | Zbl 0549.34001 · Zbl 0549.34001
[21] P. Lax , Integrals of nonlinear equations of evolution and solitary waves , Comm. Pure Appl. Math . 21 ( 1968 ), 467 - 490 . MR 235310 | Zbl 0162.41103 · Zbl 0162.41103
[22] H.P. Mckean , Integrable systems and algebraic curves , In: ” Global Analysis ”, 755 , Springer Lecture Notes in Mathematics , 1979 , pp. 83 - 200 . MR 564904 | Zbl 0449.35080 · Zbl 0449.35080
[23] P.I. Naumkin - I. Shishmarev , Nonlinear Nonlocal Equations in the Theory of Waves, vol. 133, Transl. Math. Monographs , Providence, Rhode Island , 1994 . MR 1261868 | Zbl 0802.35002 · Zbl 0802.35002
[24] A. Pazy , Semigroups of Linear Operators and Applications to Partial Differential Equations , Springer Verlag , New York , 1983 . MR 710486 | Zbl 0516.47023 · Zbl 0516.47023
[25] J. Schiff , Zero curvature formulations of dual hierarchies , J. Math. Phys. 37 ( 1996 ), 1928 - 1938 . MR 1380881 | Zbl 0863.35093 · Zbl 0863.35093
[26] E.M. Stein , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , Princeton University Press , Princeton , 1993 . MR 1232192 | Zbl 0821.42001 · Zbl 0821.42001
[27] G.B. Whitham , Linear and Nonlinear Waves , J. Wiley & Sons , New York , 1980 . MR 1699025
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