328 research outputs found
Blowup issues for a class of nonlinear dispersive wave equations
In this paper we consider the nonlinear dispersive wave equation on the real
line,
,
that for appropriate choices of the functions and includes well known
models, such as Dai's equation for the study of vibrations inside elastic rods
or the Camassa--Holm equation modelling water wave propagation in shallow
water. We establish a local-in-space blowup criterion (i.e., a criterion
involving only the properties of the data in a neighbourhood of a single
point) simplifying and extending earlier blowup criteria for this equation. Our
arguments apply both to the finite and infinite energy case, yielding the
finite time blowup of strong solutions with possibly different behavior as
and
Convergent Numerical Schemes for the Compressible Hyperelastic Rod Wave Equation
We propose a fully discretised numerical scheme for the hyperelastic rod wave
equation on the line. The convergence of the method is established. Moreover,
the scheme can handle the blow-up of the derivative which naturally occurs for
this equation. By using a time splitting integrator which preserves the
invariants of the problem, we can also show that the scheme preserves the
positivity of the energy density
Convergent numerical schemes for the compressible hyperelastic rod wave equation
We propose a fully discretised numerical scheme for the hyperelastic rod wave equation on the line. The convergence of the method is established. Moreover, the scheme can handle the blow-up of the derivative which naturally occurs for this equation. By using a time splitting integrator which preserves the invariants of the problem, we can also show that the scheme preserves the positivity of the energy densit
The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H1(R)
The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space C([0,∞)×R)nL∞([0,∞);H1(R)) under the assumption that the initial value u 0 (x) only belongs to the space H 1 (R) . The limit of viscous approximations, a one-sided super bound estimate and a space-time higher-norm estimate for the equation are established to prove the existence of the global weak solution
Wave-Breaking Phenomena and Existence of Peakons for a Generalized Compressible Elastic-Rod Equation
Consideration in this paper is the Cauchy problem of a generalized hyperelastic-rod wave equation. We first derive a wave-breaking mechanism for strong solutions, which occurs in finite time for certain initial profiles. In addition, we determine the existence of some new peaked solitary wave solutions
The H
The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the space C([0,∞) × R)∩L∞([0,∞); H1(R)) provided that its initial value u0(x) belongs to the space H1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived
On the Existence of Global Weak Solutions for a Weakly Dissipative Hyperelastic Rod Wave Equation
Assuming that the initial value v0(x) belongs to the space H1(R), we prove the existence of global weak solutions for a weakly dissipative hyperelastic rod wave equation in the space C([0,∞)×R)⋂L∞([0,∞);H1(R)). The limit of the viscous approximation for the equation is used to establish the existence
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