# zbMATH — the first resource for mathematics

Global weak solutions and blow-up structure for the Degasperis-Procesi equation. (English) Zbl 1126.35053
The authors study the equation $$u_t - u_{txx} + 4uu_{x} = 3u_{x} u_{xx} + uu_{xxx}$$ on the real line: one of several shallow water wave equations. Even though the equation is integrable (e.g., in the sense of Lax pairs), its conserved quantities do not even control the energy norm.
The authors apply the operator $$1 - \partial^2_x$$, which reduces the equation to a Burgers type evolution equation with a convolution term, and they use the conserved quantity $$\int(1 - \partial^2_x )u \cdot (4 - \partial^2_x )^{-1} u\, dx$$, which controls the $$L^2$$ norm and eventually leads to an a apriori estimate for the supremum norm.
They improve estimates of the third author to the effect that, if finite time $$T$$ blowup occurs for initial data $$u_0\in H^{ s} (s > \frac {3}{2}$$, for which local well-posedness holds), then $$\inf_x u_x \sim -1/(T - t)$$ as $$t\rightarrow T$$, whereas $$u$$ remains uniformly bounded. If $$u_0\neq 0$$ is odd and $$(1 - \partial^2_x )u_0\geq 0$$ for $$x < 0$$, then the finite blow-up occurs only at $$x = 0$$. Conversely, for $$u_0\in H^1$$ where $$(1 - \partial^2_x )u_0$$ is a Radon measure with bounded variation that is $$\leq 0$$ for $$x < x_0$$ and $$\geq 0$$ for $$x > x_0$$ (as defined in terms of support), they show the existence of global weak solutions.
They also show, despite the weak regularity hypothesis on the initial data, the uniqueness of weak solutions in this class.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35G25 Initial value problems for nonlinear higher-order PDEs 35L67 Shocks and singularities for hyperbolic equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text:
##### References:
  Beals, R.; Sattinger, D.; Szmigielski, J., Acoustic scattering and the extended korteweg – de Vries hierarchy, Adv. math., 140, 190-206, (1998) · Zbl 0919.35118  Bressan, A.; Constantin, A., Global conservative solutions of the camassa – holm equation, preprint · Zbl 1105.76013  Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 1661-1664, (1993) · Zbl 0972.35521  Camassa, R.; Holm, D.; Hyman, J., A new integrable shallow water equation, Adv. appl. mech., 31, 1-33, (1994) · Zbl 0808.76011  Coclite, G.M.; Karlsen, K.H., On the well-posedness of the degasperis – procesi equation, J. funct. anal., 233, 60-91, (2006) · Zbl 1090.35142  G.M. Coclite, K.H. Karlsen, N.H. Risebro, Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, preprint · Zbl 1246.76114  Constantin, A., Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. inst. Fourier (Grenoble), 50, 321-362, (2000) · Zbl 0944.35062  Constantin, A., On the scattering problem for the camassa – holm equation, Proc. roy. soc. London ser. A, 457, 953-970, (2001) · Zbl 0999.35065  Constantin, A., Finite propagation speed for the camassa – holm equation, J. math. phys., 46, (2005), 023506, 4 pp · Zbl 1076.35109  Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Ann. sc. norm. sup. Pisa, 26, 303-328, (1998) · Zbl 0918.35005  Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta math., 181, 229-243, (1998) · Zbl 0923.76025  Constantin, A.; Escher, J., Global weak solutions for a shallow water equation, Indiana univ. math. J., 47, 1527-1545, (1998) · Zbl 0930.35133  Constantin, A.; Kolev, B., Geodesic flow on the diffeomorphism group of the circle, Comment. math. helv., 78, 787-804, (2003) · Zbl 1037.37032  Constantin, A.; McKean, H.P., A shallow water equation on the circle, Comm. pure appl. math., 52, 949-982, (1999) · Zbl 0940.35177  Constantin, A.; Molinet, L., Global weak solutions for a shallow water equation, Comm. math. phys., 211, 45-61, (2000) · Zbl 1002.35101  Constantin, A.; Strauss, W.A., Stability of peakons, Comm. pure appl. math., 53, 603-610, (2000) · Zbl 1049.35149  Constantin, A.; Strauss, W., Stability of a class of solitary waves in compressible elastic rods, Phys. lett. A, 270, 140-148, (2000) · Zbl 1115.74339  Constantin, A.; Strauss, W.A., Stability of the camassa – holm solitons, J. nonlinear sci., 12, 415-422, (2002) · Zbl 1022.35053  Dai, H.H., Model equations for nonlinear dispersive waves in a compressible mooney – rivlin rod, Acta mech., 127, 193-207, (1998) · Zbl 0910.73036  Degasperis, A.; Holm, D.D.; Hone, A.N.W., A new integral equation with peakon solutions, Theoret. and math. phys., 133, 1463-1474, (2002)  Degasperis, A.; Procesi, M., Asymptotic integrability, (), 23-37 · Zbl 0963.35167  Drazin, P.G.; Johnson, R.S., Solitons: an introduction, (1989), Cambridge Univ. Press Cambridge · Zbl 0661.35001  Dullin, H.R.; Gottwald, G.A.; Holm, D.D., An integrable shallow water equation with linear and nonlinear dispersion, Phys. rev. lett., 87, 4501-4504, (2001)  Dullin, H.R.; Gottwald, G.A.; Holm, D.D., Camassa – holm, korteweg – de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid dynam. research, 33, 73-79, (2003) · Zbl 1032.76518  Fokas, A.; Fuchssteiner, B., Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4, 47-66, (1981) · Zbl 1194.37114  Henry, D., Infinite propagation speed for the degasperis – procesi equation, J. math. anal. appl., 311, 755-759, (2005) · Zbl 1094.35099  Holm, D.D.; Staley, M.F., Wave structure and nonlinear balances in a family of evolutionary pdes, SIAM J. appl. dyn. syst., 2, 323-380, (2003), (electronic) · Zbl 1088.76531  Johnson, R.S., Camassa – holm, korteweg – de Vries and related models for water waves, J. fluid mech., 455, 63-82, (2002) · Zbl 1037.76006  Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations, (), 25-70  Kenig, C.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized korteweg – de Vries equation via the contraction principle, Comm. pure appl. math., 46, 527-620, (1993) · Zbl 0808.35128  Lenells, J., Traveling wave solutions of the degasperis – procesi equation, J. math. anal. appl., 306, 72-82, (2005) · Zbl 1068.35163  Lenells, J., Conservation laws of the camassa – holm equation, J. phys. A, 38, 869-880, (2005) · Zbl 1076.35100  Li, Y.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. differential equations, 162, 27-63, (2000) · Zbl 0958.35119  Y. Liu, Global existence and blow-up solutions for a nonlinear shallow water equation, Math. Ann., in press · Zbl 1102.35021  Y. Liu, Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., in press  H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, preprint · Zbl 1185.35194  Lundmark, H.; Szmigielski, J., Multi-peakon solutions of the degasperis – procesi equation, Inverse problems, 19, 1241-1245, (2003) · Zbl 1041.35090  Malek, J.; Necas, J.; Rokyta, M.; Ruzicka, M., Weak and measure-valued solutions to evolutionary pdes, (1996), Chapman & Hall London · Zbl 0851.35002  Matsuno, Y., Multisoliton solutions of the degasperis – procesi equation and their peakon limit, Inverse problems, 21, 1553-1570, (2005) · Zbl 1086.35095  McKean, H.P., Integrable systems and algebraic curves, (), 83-200 · Zbl 0449.35080  Misiolek, G., A shallow water equation as a geodesic flow on the bott – virasoro group, J. geom. phys., 24, 203-208, (1998) · Zbl 0901.58022  Mustafa, O.G., A note on the degasperis – procesi equation, J. nonlinear math. phys., 12, 10-14, (2005) · Zbl 1067.35078  Natanson, I.P., Theory of functions of a real variable, (1998), Ungar New York · Zbl 0064.29102  Rodriguez-Blanco, G., On the Cauchy problem for the camassa – holm equation, Nonlinear anal., 46, 309-327, (2001) · Zbl 0980.35150  Tao, T., Low-regularity global solutions to nonlinear dispersive equations, (), 19-48 · Zbl 1042.35068  Vakhnenko, V.O.; Parkes, E.J., Periodic and solitary-wave solutions of the degasperis – procesi equation, Chaos solitons fractals, 20, 1059-1073, (2004) · Zbl 1049.35162  E. Wahlén, Global existence of weak solutions to the Camassa-Holm equation, preprint  Whitham, G.B., Linear and nonlinear waves, (1980), Wiley New York · Zbl 0373.76001  Xin, Z.; Zhang, P., On the weak solutions to a shallow water equation, Comm. pure appl. math., 53, 1411-1433, (2000) · Zbl 1048.35092  Xin, Z.; Zhang, P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. partial differential equations, 27, 1815-1844, (2002) · Zbl 1034.35115  Yin, Z., On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. math., 47, 649-666, (2003) · Zbl 1061.35142  Yin, Z., Global existence for a new periodic integrable equation, J. math. anal. appl., 283, 129-139, (2003) · Zbl 1033.35121  Yin, Z., Global weak solutions to a new periodic integrable equation with peakon solutions, J. funct. anal., 212, 182-194, (2004) · Zbl 1059.35149  Yin, Z., Global solutions to a new integrable equation with peakons, Indiana univ. math. J., 53, 1189-1210, (2004) · Zbl 1062.35094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.