Evolutions of the momentum density,deformation tensor and the nonlocal term of the Camassa–Holm equation |
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| Institution: | 2. Department of Chemistry – Ångström Laboratory, Uppsala University, Box 538, 751 21 Uppsala, Sweden;3. Applied Chemistry and Engineering Research Centre of Excellence (ACER CoE), Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid, Ben Guerir 43150, Morocco;1. Dipartimento di Scienze della Terra e dell''Ambiente, Università degli Studi di Pavia, Via Ferrata 9, 27100 Pavia, Italy;2. Department of Earth Sciences, Uppsala University, Villavägen 16, 752 36 Uppsala, Sweden;3. Dipartimento di Fisica, Università degli Studi di Pavia, Via Bassi 6, 27100 Pavia, Italy;4. Aix-Marseille University, CNRS, IRD, CEREGE, Avenue Louis Philibert, 13545 Aix en Provence, France |
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| Abstract: | Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term. |
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| Keywords: | Camassa–Holm equation Momentum density Nonlocal term Asymptotic property Blow up |
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