Single Point Gradient Blow‐Up and Nonuniqueness for a Third‐Order Nonlinear Dispersion Equation

Abstract

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 is 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) created by Oleinik in the middle of the 1950s. Contrary to and not surprisingly, self‐similar gradient blow‐up for is shown to admit a unique continuation. Bearing in mind the classic form , the NDE can be written as with the standard linear integral operator (−D2x)−1 > 0. However, because is a nonlocal equation, no standard entropy and/or BV‐approaches apply (moreover, the x‐variations of solutions of is increasing for BV data u0(x)).

Eigenfunctions and Very Singular Similarity Solutions of Odd‐Order Nonlinear Dispersion PDEs: Toward a “Nonlinear Airy Function” and Others

“…(ii) In addition, (2), as an equation with nonlinear dispersion mechanism, contains and describes other key singularity phenomena such as a complicated formation of various shock and smoother rarefaction waves, which appear from discontinuous data, as well as a general nonuniqueness of "entropy solutions" after a single point gradient blow-up. We do not touch these difficult, even still often mathematically obscure, phenomena and refer to [2,3,4] for further details.…”

“…A proper posing of "boundary conditions" for (108), to specify the corresponding Sturm-Liouville problem and next initial conditions, to get the corresponding eigenfunction expansion (110) is a difficult and uncertain problem. 4 It is also unlikely to provide us with any useful finite explicit formulae. However, analytic relationships such as (99) can promise extra asymptotic properties of the first rescaled eigenfunctionỸ 0 (y), especially the decay rate of the minimal behavior indicated in (24).…”

Shock waves and compactons for fifth-order non-linear dispersion equations

“…The situation dramatically changes if we want to treat solutions with shocks. Namely, it is known that even for the NDE-3 (1.30), the similarity formation mechanism of shocks immediately shows nonunique extensions of solutions after a typical "gradient" catastrophe [21]. Therefore, we do not have a chance to get, in such an easy (or any) manner, a uniqueness/entropy result for more complicated NDEs such as (1.5) by using the δdeformation (evolutionary smoothing) approach.…”

“…On the other hand, in a FBP setting by adding an extra suitable condition on shock lines, the problem might be well-posed with a unique solution, though proofs can be very difficult. We refer again to a more detailed discussion of these issues for the NDE-3 (1.30) in [21]. Though we must admit that, for the NDE-5 (1.5), which induces 5D dynamical systems for the similarity profiles (and hence 5D phase spaces), those nonuniqueness and non-entropy conclusions are more difficult and not that clear as for the NDEs-3, so some of their aspects do unavoidably remain questionable and even open.…”

“…Restricting the ODE (5.2) to W 6 yields an algebraic system, which admits an exact solution for the following value of the critical α c : [21] for the NDEs-3, a discontinuous shock wave extension of blow-up similarity solutions (5.1), (5.2) is assumed to be done by using the global ones (5.5), (5.6). Actually, this leads to watching a whole 5D family of solutions parameterized by their Cauchy values at the origin:…”