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17
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
On Asymptotic Variational Wave Equations
, 2004
"... We investigate the equation (ut + (f(u))x)x = f ′ ′ (u)(ux) 2 /2 where f(u) is a given smooth function. Typically f(u) = u 2 /2 or u 3 /3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation utt − c(u)(c(u)ux)x = 0 which models some liquid crystals with ..."
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Cited by 17 (9 self)
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We investigate the equation (ut + (f(u))x)x = f ′ ′ (u)(ux) 2 /2 where f(u) is a given smooth function. Typically f(u) = u 2 /2 or u 3 /3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation utt − c(u)(c(u)ux)x = 0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the EulerLagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.
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.
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
Global Conservative Solutions to a Nonlinear Variational Wave Equation
, 2008
"... 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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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.
Contractive Metrics for Nonsmooth Evolutions
, 2012
"... Abstract. Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined ..."
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Abstract. Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm. In alternative, one can construct different distance functions, related to a Riemannian structure or to an optimal transportation problem. This paper reviews various cases where this approach can be implemented, in connection with discontinuous ODEs on IR n, nonlinear wave equations, and systems of conservation laws. For all the evolution equations considered here, a metric can be constructed such that the distance between any two solutions satisfies a Gronwall type estimate. This yields the uniqueness of solutions, and estimates on their continuous dependence on the initial data.
ENERGY CONSERVATIVE SOLUTIONS TO A NONLINEAR WAVE SYSTEM OF NEMATIC LIQUID CRYSTALS
"... Abstract. We establish the global existence of solutions to the Cauchy problem for a system of hyperbolic partial differential equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and twist coefficients. Our results have no restrictions on the angles of th ..."
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Cited by 1 (1 self)
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Abstract. We establish the global existence of solutions to the Cauchy problem for a system of hyperbolic partial differential equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and twist coefficients. Our results have no restrictions on the angles of the director, as we use the director in its natural threecomponent form, rather than the twocomponent form of spherical angles. 1.
Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation
, 2015
"... The paper is concerned with conservative solutions to the nonlinear wave equation utt−c(u) c(u)ux x = 0. For an open dense set of C3 initial data, we prove that the solution is piecewise smooth in the tx plane, while the gradient ux can blow up along finitely many characteristic curves. The analysi ..."
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The paper is concerned with conservative solutions to the nonlinear wave equation utt−c(u) c(u)ux x = 0. For an open dense set of C3 initial data, we prove that the solution is piecewise smooth in the tx plane, while the gradient ux can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom’s transversality theorem. 1
Unique Conservative Solutions to a Variational Wave Equation
, 2014
"... Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation utt−c(u)(c(u)ux)x = 0. Given a solution u(t, x), even if the wave speed c(u) is only Hölder continuous in the tx plane, one can still define forward and backward character ..."
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Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation utt−c(u)(c(u)ux)x = 0. Given a solution u(t, x), even if the wave speed c(u) is only Hölder continuous in the tx plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables X,Y, constant along characteristics, we prove that t, x, u, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data u(0, ·) ∈ H1(IR), ut(0, ·) ∈ L2(IR). 1
FRG: Collaborative Research: Nonlinear Partial Differential Equations of Mixed HyperbolicElliptic Type Arising in Mechanics and Geometry
"... The project is devoted to a mathematical study of the nonlinear partial differential equations of mixed hyperbolicelliptic type arising in mechanics and geometry. Such issues occur naturally in both multidimensional gas dynamics and isometric embedding problems in differential geometry. The propose ..."
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The project is devoted to a mathematical study of the nonlinear partial differential equations of mixed hyperbolicelliptic type arising in mechanics and geometry. Such issues occur naturally in both multidimensional gas dynamics and isometric embedding problems in differential geometry. The proposers will collaborate in a three year effort to advance the mathematical understanding of these equations, their applications, and numerical resolution. Success in this project will advance knowledge of this fundamental area of mathematics and mechanics and will introduce a new generation of researchers to the outstanding problems in the field. Intellectual Merit: The mathematical issues to be addressed are fundamental for the understanding of systems of partial differential equations of mixed type. Furthermore, since the design of efficient numerical schemes hinges on the understanding of the underlying mathematical structure, success in this project will be useful to numerical analysis. Broader Impact: The project will (1) yield new understanding of the mathematics of geometry, fluids, and gases, that is critical for aerodynamics, computer sciences, medical imaging, industrial gas processing, and environmental science; (2) include outstanding female scientists, members from underrepresented groups as PIs, consultants,