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Global conservative solutions of the CamassaHolm equation
"... Abstract. This paper develops a new approach in the analysis of the CamassaHolm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new va ..."
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Cited by 90 (7 self)
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Abstract. This paper develops a new approach in the analysis of the CamassaHolm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time. The nonlinear partial differential equation ut − utxx + 3uux = 2uxuxx + uuxxx, t> 0, x ∈ IR, was derived by Camassa and Holm [CH] as a model for the propagation of shallow water waves, with
Global solutions of the HunterSaxton equation
 SIAM J. Math. Anal
"... Abstract. We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respec ..."
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Cited by 40 (5 self)
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Abstract. We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.
Global dissipative solutions of the CamassaHolm equation
 Anal. Appl. (Singap
"... Abstract. This paper is devoted to the continuation of solutions to the CamassaHolm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L ∞ space, containing a nonlocal source term ..."
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Cited by 38 (3 self)
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Abstract. This paper is devoted to the continuation of solutions to the CamassaHolm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L ∞ space, containing a nonlocal source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data ū ∈ H 1 (IR), and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking. The CamassaHolm equation ut − utxx + 3uux = 2uxuxx + uuxxx t> 0, x ∈ IR, (0.1) models the propagation of water waves in the shallow water regime, when the wavelength is considerably larger than the average water depth. Here u(t, x) represents the water’s free surface over
Weak solutions to a nonlinear variational wave equation with general data
 Annals of Inst. H. Poincaré
"... We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation utt − c(u)(c(u)ux)x = 0, for initial data of finite energy. Here c(·) is any smooth function with uniformly positive bounded values. ..."
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Cited by 17 (9 self)
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We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation utt − c(u)(c(u)ux)x = 0, for initial data of finite energy. Here c(·) is any smooth function with uniformly positive bounded values.
An Optimal Transportation Metric for Solutions of the CamassaHolm Equation
"... Dedicated to Prof. Joel Smoller in the occasion of his 65th birthday Abstract. In this paper we construct a global, continuous flow of solutions to the CamassaHolm equation on the entire space H1 ∫. Our solutions are conservative, in the sense that the total energy 2 2 (u + ux) dx remains a.e. con ..."
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Cited by 16 (5 self)
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Dedicated to Prof. Joel Smoller in the occasion of his 65th birthday Abstract. In this paper we construct a global, continuous flow of solutions to the CamassaHolm equation on the entire space H1 ∫. Our solutions are conservative, in the sense that the total energy 2 2 (u + ux) dx remains a.e. constant in time. Our new approach is based on a distance functional J(u, v), defined in terms of an optimal transportation problem, which satisfies d dtJ(u(t), v(t)) ≤ κ · J(u(t), v(t)) for every couple of solutions. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multipeakon solutions, and prove a general uniqueness result. The CamassaHolm equation can be written as a scalar conservation law with an additional integrodifferential term: ut + (u 2 /2)x + Px = 0, (1.1)
Lipschitz metric for the Hunter–Saxton equation
, 2009
"... We study stability of solutions of the Cauchy problem for the Hunter–Saxton equation ut + uux = 1 4 (R x − ∞ u2x dx − R ∞ x u2x dx) with initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t), v(t)) ≤ e C ..."
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Cited by 14 (7 self)
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We study stability of solutions of the Cauchy problem for the Hunter–Saxton equation ut + uux = 1 4 (R x − ∞ u2x dx − R ∞ x u2x dx) with initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t), v(t)) ≤ e Ct dD(u0, v0).
CONVERGENT DIFFERENCE SCHEMES FOR THE HUNTER–SAXTON EQUATION
"... Abstract. We propose and analyze several finite difference schemes for the Hunter–Saxton equation (HS) ut + uux = 1 ∫ x (ux) 2 0 2 dx, x> 0, t>0. This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique di ..."
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Cited by 4 (4 self)
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Abstract. We propose and analyze several finite difference schemes for the Hunter–Saxton equation (HS) ut + uux = 1 ∫ x (ux) 2 0 2 dx, x> 0, t>0. This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results. 1.
Orientation waves in a director field with rotational inertia, Kinet
 Relat. Models
"... (Communicated by Pierangelo Marcati) Abstract. We study a variational system of nonlinear hyperbolic partial differential equations that describes the propagation of orientation waves in a director field with rotational inertia and potential energy given by the OseenFrank energy from the continuum ..."
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Cited by 4 (0 self)
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(Communicated by Pierangelo Marcati) Abstract. We study a variational system of nonlinear hyperbolic partial differential equations that describes the propagation of orientation waves in a director field with rotational inertia and potential energy given by the OseenFrank energy from the continuum theory of nematic liquid crystals. There are two types of waves, which we call splay and twist waves, respectively. Weakly nonlinear splay waves are described by the quadratically nonlinear HunterSaxton equation. In this paper, we derive a new cubically nonlinear asymptotic equation that describes weakly nonlinear twist waves. This equation provides a surprising representation of the HunterSaxton equation, and like the HunterSaxton equation it is completely integrable. There are analogous cubically nonlinear representations of the CamassaHolm and DegasperisProcesi equations. Moreover, two different, but compatible, variational principles for the HunterSaxton equation arise from a single variational principle for the primitive director field equations in the two different limits for splay and twist waves. We also use the asymptotic equation to analyze a onedimensional initial value problem for the directorfield equations with twistwave initial data.
Erosion Profile by a Global Model for Granular Flow
 ARMA
, 2010
"... In this paper we study an integrodifferential equation that models the erosion of a mountain profile caused by small avalanches. The equation is in conservative form, with a nonlocal flux involving an integral of the mountain slope. Under suitable assumptions on the erosion rate, the mountain prof ..."
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Cited by 3 (3 self)
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In this paper we study an integrodifferential equation that models the erosion of a mountain profile caused by small avalanches. The equation is in conservative form, with a nonlocal flux involving an integral of the mountain slope. Under suitable assumptions on the erosion rate, the mountain profile develops several types of singularities, which we call kinks, shocks and hyperkinks. We study formation of these singularities, and derive admissibility conditions. Furthermore, entropy weak solutions to the Cauchy problem are constructed globally in time, taking limits of piecewise affine approximate solutions. Entropy and entropy flux functions are introduced, and Lax entropy condition is established for the weak solutions.
Representation of dissipative solutions to a nonlinear variational wave equation
 Comm. Math. Sci
"... The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuou ..."
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Cited by 2 (2 self)
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The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov’s compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy is a monotone decreasing function of time. 1