# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Lipschitz metric for the Hunter-Saxton equation. (English) Zbl 1195.35046
Summary: We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation $u_t + uu_x = \frac 14 (\int^x_{-\infty} u^2_x dx)$ with initial data $u_{0}$. In particular, we derive a new Lipschitz metric $d_{\cal D}$ with the property that for two solutions $u$ and $v$ of the equation we have $d_{\cal D} (u(t),v(t)) \leqslant e^{C^t} d_{\cal D} (u_0,v_0)$.

##### MSC:
 35B35 Stability of solutions of PDE 35R09 Integro-partial differential equations
Full Text:
##### References:
 [1] Hunter, J. K.; Saxton, R.: Dynamics of director fields, SIAM J. Appl. math. 51, No. 6, 1498-1521 (1991) · Zbl 0761.35063 · doi:10.1137/0151075 [2] Hunter, J. K.; Zheng, Y. X.: On a completely integrable nonlinear hyperbolic variational equation, Phys. D 79, No. 2 -- 4, 361-386 (1994) · Zbl 0900.35387 [3] Hunter, J. K.; Zheng, Y. X.: On a nonlinear hyperbolic variational equation. I. global existence of weak solutions, Arch. ration. Mech. anal. 129, No. 4, 305-353 (1995) · Zbl 0834.35085 · doi:10.1007/BF00379259 [4] Hunter, J. K.; Zheng, Y. X.: On a nonlinear hyperbolic variational equation. II. the zero-viscosity and dispersion limits, Arch. ration. Mech. anal. 129, No. 4, 355-383 (1995) · Zbl 0834.35085 [5] Zhang, P.; Zheng, Y.: On oscillations of an asymptotic equation of a nonlinear variational wave equation, Asymptot. anal. 18, No. 3 -- 4, 307-327 (1998) · Zbl 0935.35031 [6] Zhang, P.; Zheng, Y.: On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation, Acta math. Sin. (Engl. Ser.) 15, No. 1, 115-130 (1999) · Zbl 0930.35142 · doi:10.1007/s10114-999-0063-7 [7] Zhang, P.; Zheng, Y.: Existence uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data, Arch. ration. Mech. anal. 155, No. 1, 49-83 (2000) · Zbl 0982.35062 · doi:10.1007/s002050000107 [8] Bressan, A.; Constantin, A.: Global solutions of the Hunter -- Saxton equation, SIAM J. Math. anal. 37, No. 3, 996-1026 (2005) · Zbl 1108.35024 · doi:10.1137/050623036 [9] Holden, H.; Karlsen, K. H.; Risebro, N. H.: Convergent difference schemes for the Hunter -- Saxton equation, Math. comp. 76, 699-744 (2007) · Zbl 1114.65101 · doi:10.1090/S0025-5718-07-01919-9 [10] Bressan, A.; Zhang, P.; Zheng, Y.: Asymptotic variational wave equations, Arch. ration. Mech. anal. 183, No. 1, 163-185 (2007) · Zbl 1168.35026 · doi:10.1007/s00205-006-0014-8 [11] Bressan, A.; Fonte, M.: An optimal transportation metric for solutions of the Camassa -- Holm equation, Methods appl. Anal. 12, 191-220 (2005) · Zbl 1133.35054 [12] Kružkov, S. N.: First order quasilinear equations with several independent variables, Mat. sb. (N.S.) 81, No. 123, 228-255 (1970) [13] Bressan, A.: Hyperbolic systems of conservation laws, (2000) · Zbl 0987.35105 [14] Holden, H.; Risebro, N. H.: Front tracking for hyperbolic conservation laws, (2007) · Zbl 1239.35003 [15] Bressan, A.; Colombo, G.: Existence and continuous dependence for discontinuous o.d.e.s, Boll. un. Mat. ital. B (7) 4, No. 2, 295-311 (1990) · Zbl 0709.34003 [16] Holden, H.; Raynaud, X.: Global conservative solutions of the Camassa -- Holm equation --- a Lagrangian point of view, Comm. partial differential equations 32, No. 10 -- 12, 1511-1549 (2007) · Zbl 1136.35080 · doi:10.1080/03605300601088674 [17] Bressan, A.; Constantin, A.: Global conservative solutions of the Camassa -- Holm equation, Arch. ration. Mech. anal. 183, No. 2, 215-239 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z