Global Weak Solutions for a Shallow Water Equation

By Giuseppe Maria Coclite, Helge Holden, Kenneth Hvistendahl Karlsen. 2008

268 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/947a1dfcc71d1a7e3f84cd6547c800e62ad88d6c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Global Conservative Solutions of the Camassa–Holm Equation (2007, background): “They have to be understood [11] as weak solutions of the Camassa–Holm equation, for which purpose it is suitable to recast (1.”
Global weak solutions for a two-component Camassa–Holm shallow water system (2011, methodology): “On the other hand, it has global weak solutions in H 1(R) [2,3,15,22,47].”
Initial Boundary Value Problems of the Camassa–Holm Equation (2008, background): “On the other hand, it has global weak solutions in H1 I , cf. Bressan and Constantin (2007), Constantin and Molinet (2000), Danchin (2001), Molinet (2004), Wahlén (2006), and Xin and Zhang (2000).”
Stability of multipeakons (2008, unknown): “In Section 2 we state a well-posedness result for (C–H) established in [13] and [16].”
Global conservative solutions of the generalized hyperelastic-rod wave equation ✩ (2007, background): “Specifically we mention that Constantin, Escher, and Molinet [8, 9] showed the following result: If the initial data u|t=0 = ū ∈ H(1)(R) and m̄ := ū − ū′′ is a positive Radon measure, then equation (1.”
Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View (2007, background): “More precisely, Constantin, Escher, and Molinet (Constantin and Escher, 1998a,b; Constantin and Molinet, 2000) showed the following result: If the initial data u t=0 = ū ∈ H1 and m̄ = ū− ū′′ is a positive Radon measure, then equation (1.3) has a unique global weak solution u ∈ C 0 T H1 , for any…”
Dissipative solutions for the Camassa-Holm equation (2009, background): “More precisely, Constantin, Escher, and Molinet [12] [14] [15] showed the following result: If the initial data u| t=0 = ¯ u ∈ H 1 (R) and 2 HOLDEN AND RAYNAUD ¯ u − ¯ u is a positive Radon measure, then equation (1.2) has a unique global weak solution u ∈ C([0, T ], H 1 (R)), for any T positive,…”
ON THE UNIQUENESS OF DISCONTINUOUS SOLUTIONS TO THE DEGASPERIS–PROCESI EQUATION (2007, background): “All solutions encompassed by Yin’s well-posedness theory are regular, that is, they are no worse than H 1, a fact that is reminiscent of the Camassa–Holm equation (see for example [5]).”
Low-Regularity Solutions of the Periodic Camassa–Holm Equation (2007, background): “For such s and p (at least in the case s > 3/2, p = 2), Theorem 1 is well known—see e.g., Constantin and Escher (1998a), Constantin and Molinet (2000), Danchin (2001), Himonas and Misiolek (2000), Li and Olver (2000), and Misiolek (2002).”
Well-posedness of modified Camassa-Holm equations (2009, methodology): “3 in [7] applies here with little change, and we omit it.”
On the well-posedness of the Degasperis-Procesi equation (2006, background): “Existence and uniqueness results for global weak solutions of the Camassa–Holm equation are proved in [4,9,12,34,35,15,16].”
A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation (2006, methodology): “Constantin, A., Escher, J. (1998c).”
Global weak solutions for the Novikov equation (2011, background): “In section 3, as in [9, 30], we show the existence and uniqueness of global weak solutions to equation (1.”