Article

Computational Mathematics and Mathematical Physics

, Volume 48, Issue 10, pp 1784-1810

First online:

Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves

  • V. A. GalaktionovAffiliated withDepartment of Mathematical Sciences, University of Bath Email author 
  • , S. I. PohozaevAffiliated withSteklov Institute of Mathematics, Russian Academy of Sciences

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Abstract

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation in
$$ u_t = (uu_x )_{xx} in\mathbb{R} \times \mathbb{R}_ + . $$
(0.1)
It is shown that two basic Riemann problems for Eq. (0.1) with the initial data
$$ S_ \mp (x) = \mp \operatorname{sgn} x $$
exhibit a shock wave (u(x, t) ≡ S (x)) and a smooth rarefaction wave (for S +), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u t + uu x = 0, which was developed by O.A. Oleinik and S.N. Kruzhkov (for equations in x ∊ ℝ N ) in the 1950s–1960s.

Keywords

general theory of partial differential equations nonlinear dispersive equations shock waves rarefaction and blowup waves Riemann’s problem entropy theory of scalar conservation laws