Stability of traveling waves for the Burgers-Hilbert equation

We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation $\delta$ from a global periodic traveling wave with small amplitude $\epsilon$. We use a modified energy method to prove the existence time of smooth solutions on a time scale of $\frac{1}{\epsilon\delta}$ with $0<\delta\ll\epsilon\ll1$ and on a time scale of $\frac{\epsilon}{\delta^2}$ with $0<\delta\ll\epsilon^2\ll1$. Moreover, we show that the traveling wave exists for an amplitude $\epsilon$ in the range $(0,\epsilon^*)$ with $\epsilon^*\sim 0.29$ and fails to exist for $\epsilon>\frac{2}{e}$.Comment: 57 pages, 1 figur

A nonlocal model describing tumor angiogenesis

In this paper, we derive and study a new mathematical model that describes the onset of angiogenesis. This new model takes the form of a nonlocal Burgers equation with both diffusive and dispersive terms. For a particular value of the parameters, the equation reduces to ∂tp −1/2(−Δ)(α−1)/2H∂tp = −1/2(−Δ)α/2p + p∂xp − ∂xp, where H denotes the Hilbert transform. In addition to the derivation of the new model, the main novelty of the present paper is that we also prove a number of well-posedness results. Finally, some preliminary numerical results are shown. These numerical results suggest that the dynamics of the equation is rich enough to have solutions that blow up in finite time.R.G-B was supported by the project “Mathematical Analysis of Fluids and Applications”, Spain Grant PID2019-109348GA-I00 funded by MCIN/AEI/, Spain 10.13039/501100011033 and acronym “MAFyA”. This publication is part of the project PID2019-109348GA-I00/AEI/10.13039/501100011033. R.G-B is also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation, Spain. The BBVA Foundation accepts no responsibility for the opinions, statements, and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors