Abstract
We consider two different non-local, and non-linear transport equations, both of which form singularities in finite time, starting from smooth initial conditions. The first,
θt=D(γ)(θ)θx,
is a non-local version of the inviscid Burgers' equation, which is hyperbolicand forms a shock in finite time; D(γ) denotes the fractional derivative, which for γ=0 is the Hilbert transform: D(0)(θ)=H(θ). We show that singular solutions of the non-local equation for γ<1 connect to the hierarchy of shock solutions of Burgers' equation, which are obtained for γ<1. The second equation,
θt − δ(θHθ)x − (1 − δ)(Hθ)θx = 0,
is a simplified version of a class of ill-posed problems arising in the theory of vortex sheets and water waves, which are known to exhibit a weak curvature singularity in finite time, known as "Moore's singularity''. The linearized form of (2) allows for a continuous family of curvature singularities, with the scaling exponent α as a parameter, each of which is identical to those arising in Moore's singularity. By considering the stability of each singularity, we are able to determine which exponent is selected, and show that its value depends on the parameter δ.
θt=D(γ)(θ)θx,
is a non-local version of the inviscid Burgers' equation, which is hyperbolicand forms a shock in finite time; D(γ) denotes the fractional derivative, which for γ=0 is the Hilbert transform: D(0)(θ)=H(θ). We show that singular solutions of the non-local equation for γ<1 connect to the hierarchy of shock solutions of Burgers' equation, which are obtained for γ<1. The second equation,
θt − δ(θHθ)x − (1 − δ)(Hθ)θx = 0,
is a simplified version of a class of ill-posed problems arising in the theory of vortex sheets and water waves, which are known to exhibit a weak curvature singularity in finite time, known as "Moore's singularity''. The linearized form of (2) allows for a continuous family of curvature singularities, with the scaling exponent α as a parameter, each of which is identical to those arising in Moore's singularity. By considering the stability of each singularity, we are able to determine which exponent is selected, and show that its value depends on the parameter δ.
| Original language | English |
|---|---|
| Pages (from-to) | 325-340 |
| Number of pages | 16 |
| Journal | Nonlinearity |
| Volume | 33 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2 Dec 2019 |
Keywords
- nonlocal transport equations
- similarity solutions
- similarity of the second kind