Blow-up analysis and spatial asymptotic profiles of solutions to a modified two-component hyperelastic rod system
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Blow-up analysis and spatial asymptotic profiles of solutions to a modified two-component hyperelastic rod system Long Wei1
· Qi Zeng1
Received: 20 March 2019 / Revised: 20 March 2019 / Accepted: 10 November 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we consider a modified two-component hyperelastic rod system, which is a generalization of the Camassa–Holm equation modeling shallow water waves moving over a linear shear flow. We show blow-up analysis and exact spatial asymptotic profiles of solutions to the system. Especially, for the special case γ = 1, we present the exact blow-up rate of the breaking-wave solution. Keywords Two-component hyperelastic rod system · Blow-up criterion · Blow-up rate · Persistence property Mathematics Subject Classification Primary 35Q35 · 35B44
1 Introduction In this paper,we consider a modified two-component hyperelastic rod system [24] u t − u x xt + 3uu x − γ (2u x u x x + uu x x x ) + ρ ρ¯x = 0, t > 0, x ∈ R, t > 0, x ∈ R, ρt + (ρu)x = 0,
(1.1)
where γ ∈ R is a constant, u denotes velocity field, ρ = (1 − ∂x2 )(ρ¯ − ρ¯0 ) is pointwise density (or free surface elevation from equilibrium), ρ¯ is locally averaged density and ρ¯0 is taken to be a constant.
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Long Wei [email protected] Qi Zeng [email protected]
1
Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, Zhejiang, China 0123456789().: V,-vol
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Page 2 of 15
L. Wei, Q. Zeng
Obviously, if ρ = 0, the system (1.1) is reduced to the well-known rod equation u t − u x xt + 3uu x − γ (2u x u x x + uu x x x ) = 0,
(1.2)
which is a typical model equation for an infinitely long rod composed of a general compressible hyperelastic material, which is derived by Dai in [15,16]. Especially, for γ = 0, the rod Eq. (1.2) reduces to the well-known BBM equation, which models the motion of internal gravity waves in shallow channel [3]. All solutions of the equation are global, and the solitary waves are smooth and orbitally stable [36]. For γ = 1, (1.2) is the famous Camassa–Holm (CH) equation, which models the unidirectional propagation of shallow water waves over a flat bottom [5,14,29], and water waves moving over an underlying shear flow [31]. It also arises in the study of a certain non-Newtonian fluids [4]. The CH equation has many remarkable properties: a bi-Hamiltonian structure, Lax completely integrability, infinitely many conservation laws, peakons, wave breaking, etc. [7–9,11,34,44]. The CH equation is thus much better understood than Eq. (1.2). We refer to some useful survey papers on such equation [10,35] and the references therein. Recently, Brandolese [2] established a new blow-up criterion for (1.2). The Eq. (1.2) has been generalized to many integrable multi-components systems. One of the most popular generalizations is u t − u x xt + 3uu x − γ (2u x u x x + uu x x x ) + ρρx = 0, t > 0, x ∈ R, t > 0, x ∈ R. ρt + (ρu)x = 0,
(1.3)
For γ = 1, (1.3) is the famous 2CH system and has been studied widely. The local wellposedness for the system with initial data (u 0 , ρ0 )
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