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 $$L^1({{\mathbb {R}}}^n)$$ L 1 ( R n ) , and more generally in $$L^p({{\mathbb {R}}}^n)$$ L p ( R n ) . We investigate here the situation where u (0) is a bounded measure instead, focusing on the case $$n=2$$ n = 2 . This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as $$t\rightarrow +\infty $$ t → + ∞ .

中文翻译：

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多维 Burgers 方程的源解

我们在最近的一次合作中表明，当初始数据 u (0) 在 Lebesgue 空间 $$L^1({{\mathbb {R}} }^n)$$ L 1 ( R n ) ，更一般地在 $$L^p({{\mathbb {R}}}^n)$$ L p ( R n ) 中。我们在这里调查 u (0) 是有界测度的情况，重点关注 $$n=2$$n = 2 的情况。这是由具有可积数据的解决方案的渐近行为的描述所激发的，如 $$t\rightarrow +\infty $$ t → + ∞ 。