Premium
Single Point Gradient Blow‐Up and Nonuniqueness for a Third‐Order Nonlinear Dispersion Equation
Author(s) -
Galaktionov Victor A.
Publication year - 2011
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2010.00499.x
Subject(s) - mathematics , uniqueness , mathematical analysis , nonlinear system , bounded function , initial value problem , conservation law , cauchy problem , partial differential equation , operator (biology) , entropy (arrow of time) , euler equations , physics , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene
A basic mechanism of a formation of shocks via gradient blow‐up from analytic solutions for the third‐order nonlinear dispersion PDE from compacton theory 1is studied. Various self‐similar solutions exhibiting single point gradient blow‐up in finite time, as t → T − < ∞ , with locally bounded final time profiles u ( x , T − ) , are constructed. These are shown to admit infinitely many discontinuous shock‐type similarity extensions for t > T , all of them satisfying generalized Rankine–Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow‐up is detected. This is in striking difference with general uniqueness‐entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics) 2created by Oleinik in the middle of the 1950s. Contrary to (1) and not surprisingly, self‐similar gradient blow‐up for (2) is shown to admit a unique continuation. Bearing in mind the classic form (2), the NDE (1) can be written as 3with the standard linear integral operator (− D 2 x ) −1 > 0 . However, because (3) is a nonlocal equation, no standard entropy and/or BV ‐approaches apply (moreover, the x ‐variations of solutions of (3) is increasing for BV data u 0 ( x ) ).