Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates

Abstract : The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data u 0 is a bounded measurable function (Kruzhkov). The semi-group (S t) t≥0 is contracting in the L 1-distance. For the multi-dimensional Burgers equation, we show that (S t) t≥0 extends uniquely as a continuous semi-group over L p (R n) whenever 1 ≤ p < ∞, and u(t) := S t u 0 is actually an entropy solution to the Cauchy problem. When p ≤ q ≤ ∞ and t > 0, S t actually maps L p (R n) into L q (R n). These results are based upon new dispersive estimates. The ingredients are on the one hand Compensated Integrability, and on the other hand a De Giorgi-type iteration.
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Submitted on : Saturday, August 18, 2018 - 10:24:07 AM
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Denis Serre, Luis Silvestre. Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates. 2018. ⟨ensl-01858016⟩

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