Weak solutions to a nonlinear variational wave equation and some related problems

Zhang, Ping

doi:10.1007/s10778-006-0111-2

Cited by 8 publications

(3 citation statements)

References 42 publications

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“…In [15,16], we dealt with a one-dimensional variational wave system arising from the variational principle (1.1) and constructed a global energy-conservative weak solution to its initial value problem. For more related papers, see [8,9,17,22,23] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Conservative solutions to a one-dimensional nonlinear variational wave equation

2015

Journal of Differential Equations

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This paper is focused on a nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle whose action is a quadratic function of the derivatives of the field. We establish the global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy. The approach follows very closely the method of energy-dependent coordinates proposed by Bressan, Zhang and Zheng [6,7]. By introducing a new set of variables, which resolve all singularities due to the possible concentration of energy, the equation can be rewritten as a semilinear system. We construct the global weak solution by expressing the solution of the semilinear system in terms of the original variables.

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Section: Introductionmentioning

confidence: 99%

Conservative solutions to a one-dimensional nonlinear variational wave equation

2015

Journal of Differential Equations

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“…The existence of a dissipative weak solution to the initial value problem of (1.6) has been extensively studied by Zhang and Zheng [8][9][10][11][12]. In [13,14], Bressan et al established an energy-conservative weak solution to the Cauchy problem of (1.6) and its asymptotic equation.…”

Section: Introductionmentioning

confidence: 99%

Global energy conservative solutions to a system of variational wave equations

2012

Nonlinear Analysis: Theory, Methods & Applications

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“…However, the a priori estimates only give the weak compactness. To overcome this difficulty, we adopt the Young measure theory spirited by the works of [13,15,[18][19][20]. Based on a one-sided supernorm estimate (see (2.4)), we show that the Young measure associated with the sequence {v ε } deduces to a Dirac measure, which gives the strong compactness of the approximations in L q loc (R + , L r loc (R + )) for q < ∞ and r < 2.…”

Section: Introductionmentioning

confidence: 99%