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<p>We establish that, in general, there are no entropy solutions to the Cauchy problem for the multidimensional Burgers equation when the initial data is a measure. This answers in the negative a conjecture of <span>D. Serre–L. Silvestre</span> (Arch Rat Mech Anal 234:1391–1411, 2019). This conjecture was motivated by the description of the asymptotic behavior of entropy solutions with integrable initial data. Despite this negative result, we still derive some information on the asymptotic behavior of such solutions. Our method relies on new Laplace transform estimates for the propagation of information.</p>

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On the Existence of Source-Solutions to the Multidimensional Burgers Equation

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年5月7日

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  • João Fernando Nariyoshi

摘要

We establish that, in general, there are no entropy solutions to the Cauchy problem for the multidimensional Burgers equation when the initial data is a measure. This answers in the negative a conjecture of D. Serre–L. Silvestre (Arch Rat Mech Anal 234:1391–1411, 2019). This conjecture was motivated by the description of the asymptotic behavior of entropy solutions with integrable initial data. Despite this negative result, we still derive some information on the asymptotic behavior of such solutions. Our method relies on new Laplace transform estimates for the propagation of information.