Abstract
We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space \(L^1({{\mathbb {R}}}^n)\), and more generally in \(L^p({{\mathbb {R}}}^n)\). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case \(n=2\). This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as \(t\rightarrow +\infty \).
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Notes
Here the divergence must be taken in the time and space variables.
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Acknowledgements
The author thanks Alberto Bressan and Luis Silvestre for valuable discussions. He is grateful to the anonymous referees for their careful reading and valuable suggestions. Part of this research was done during a stay at the Department of Mathematics of Pennsylvania State University.
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Communicated by I. Fonseca
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Serre, D. Source-Solutions for the Multi-dimensional Burgers Equation. Arch Rational Mech Anal 239, 95–116 (2021). https://doi.org/10.1007/s00205-020-01576-6
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DOI: https://doi.org/10.1007/s00205-020-01576-6