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The study reveals no entropy solutions for the Cauchy problem in multidimensional Burgers equation, suggesting new directions for research.
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.
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João F. C. Nariyoshi (2026) studied this question.
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