## Abstract

We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order \(\alpha \), and the fractal Burgers equation of order \(\beta \), where \(\alpha , \beta \in [0,1)\), and the Whitham equation. For all \(\alpha , \beta \in [0,1)\), we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a “shock-like” singularity. Moreover, we provide an asymptotic description of the blow-up. To the best of our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range \(\alpha \in [2/3, 1)\), as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range \(\beta \in [2/3, 1)\). Our construction is based on modulation theory, where the well-known smooth self-similar solutions to the inviscid Burgers equation are used as profiles. A somewhat amusing point is that the profiles that are less stable under initial data perturbations (in that the number of unstable directions is larger) are more stable under perturbations of the equation (in that higher order dispersive and/or dissipative terms are allowed) due to their slower rates of concentration. Another innovation of this article, which may be of independent interest, is the development of a streamlined weighted \(L^{2}\)-based approach (in lieu of the characteristic method) for establishing the sharp spatial behavior of the solution in self-similar variables, which leads to the sharp Hölder regularity of the solution up to the blow-up time.

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## Notes

Here, (

*s*,*y*,*U*) are the self-similar variables for solutions defined for negatives times that blow up at \((t, x) = (0, 0)\).It is important to distinguish the perturbations of the equations discussed here, which are terms of the form

*Lu*for some linear operator*L*, with an external forcing term*f*, which is independent of*u*. The effect of an external forcing term with compact support in spacetime should resemble that of a compactly supported initial data perturbation.This property is equivalent to the requirement that the Fourier multiplier \(\Gamma \partial _{x} + \Upsilon \) maps real-valued functions to real-valued functions.

More precisely, we will have boundedness of the gradient of these terms, and controlled growth for the terms themselves.

For this purpose, we need to ensure that the coefficient \((2k)! + w_{2k+1}\) is uniformly bounded away from zero; this assertion will be one of the bootstrap assumptions below.

Local well-posedness of the equation considered holds, for instance, in the space \(H^{2k+3}(\mathbb {R})\): if \(u_0 \in H^{2k+3}(\mathbb {R})\), there exists a local-in-time classical solution \(u(t,x)\in C^1([0,T], H^{2k+3}(\mathbb {R}))\) which solves the Eq. (1), and such that \(u(0,x) = u_0(x)\).

The reason why we separate out the case is \(\max \{\alpha , \beta \} = \frac{1}{2k+1}\) is entirely technical; see Lemma 5.1 below. We note that \(\frac{2k-\frac{3}{2}}{2k}\) can be replaced by any positive number strictly less than \(\frac{2k-1}{2k}\).

Recall that \(P_{\le 0}\) is the Fourier multiplier operator with symbol \(P_{\le 0}(\xi )\), where \(P_{\le 0}(\xi )\) is a nonnegative smooth function supported in \([-2, 2]\) which equals 1 on \([-1, 1]\). Moreover, \(P_{> 0}(\xi ) = 1 - P_{\le 0}(\xi )\).

Recall that \({\tilde{P}}_k\) is a function whose support properties and bounds are the same as \(P_k\).

Recall that \(\vert {\tilde{K}_{k}(y)}\vert \lesssim _{N} \frac{2^{k}}{\langle {2^{k} y}\rangle ^{N}}\).

Note that, for the nonlinear ODE \(\dot{x} = -x - x^2\), the equilibrium point \(x = 0\) attracts all orbits originating in \((-1, \infty )\).

This shows a lower bound. The upper bound comes from (68).

Note that this choice can be made independently of \(\sigma _1\).

This follows from bound (64) on \(E^{(1)}\).

Recall display (25), where

*D*and*M*are defined, and moreover recall that \({\mathcal {N}}(\vec {w}(s))\) is a vector with quadratic entries as functions of the entries of \(\vec {w}\), and \(\vec {f}\) is the vector \(((1+e^{s} \tau _{s}) F^{(2)}(s,0), \ldots , (1+e^{s} \tau _{s}) F^{(2k-1)}(s, 0))\).We use Lipschitz continuity here as it is easier to observe for (1), which is quasilinear. The minor price we have to pay is that we cannot work with the highest order topology \(H^{2k+3}\), but rather with the lower order topology \(H^{2k+2}\).

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## Acknowledgements

F. Pasqualotto would like to acknowledge Tristan Buckmaster and Javier Gómez-Serrano for insightful discussions on blow-up constructions. This material is based partially upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2021 semester. S.-J. Oh was partially supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1702-02, a Sloan Research Fellowship and a National Science Foundation CAREER Grant under NSF-DMS-1945615.

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Oh, SJ., Pasqualotto, F. Gradient Blow-Up for Dispersive and Dissipative Perturbations of the Burgers Equation.
*Arch Rational Mech Anal* **248**, 54 (2024). https://doi.org/10.1007/s00205-024-01985-x

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DOI: https://doi.org/10.1007/s00205-024-01985-x