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Third-order nonlinear dispersive equations: shocks, rarefaction, and blowup waves. (Russian, English) Zbl 1177.76183
Zh. Vychisl. Mat. Mat. Fiz. 48, No. 10, 1819-1846 (2008); translation in Comput. Math. Math. Phys. 48, No. 10, 1784-1810 (2008).
Summary: Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation \[ u_t = (uu_x)_{xx}\quad \text{in } \mathbb R \times \mathbb R_+. \tag{1} \] It is shown that two basic Riemann problems for (1) with the initial data \(S_{\mp}(x) =\mp\text{sgn} x\) exhibit a shock wave \((u(x,t)\equiv S(x))\) and a smooth rarefaction wave (for \(S_+\)), respectively. Various blowing-up and global similarity solutions to (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 (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 \(\mathbb R^N\)) in the 1950s–1960s.

MSC:
76L05 Shock waves and blast waves in fluid mechanics
35L67 Shocks and singularities for hyperbolic equations
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