Preprint

Inserted: 19 oct 2026
Last Updated: 19 oct 2026

Year: 2026

Abstract:

We study $L^2$-type contraction properties, up to a spatial shift, for a nonlocal Burgers equation with a one-sided exponential kernel. For a shock connecting a positive state to vacuum, we exhibit an explicit two-bump perturbation of the initial datum for which no Lipschitz shift starting at the initial shock position makes the squared $L^2$ distance between the averaged densities non-increasing, even for short times. We also identify two classes of initial data that admit a contractive shift locally in time, and construct data for which such a shift exists locally but cannot be continued to all positive times. For small shocks connecting two positive states, we prove that a logarithmic relative entropy is globally non-increasing along a suitable shift, for all initial data that take values between the two end states and differ from the traveling wave by a function in $L^1(\mathbb{R})$. The resulting estimates control both the distance to the local entropy shock and the size of the shift. Finally, near a Lipschitz solution of the local equation, and for identical initial data, we obtain an $O(\sqrt{\varepsilon})$ convergence rate locally in $L^2$: for both the density and its average in the case of the exponential kernel, and for the average alone in the case of general monotone kernels satisfying a quantitative decay condition.

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