# On a Slightly Compressible Water Wave

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Chenyun Luo , Johns Hopkins University
Fine Hall 322

In this talk, I would like to go over some recent results on a compressible water wave. We generalize the apriori energy estimates for the compressible Euler equations established in Lindblad-Luo to when the fluid domain is unbounded. In addition, we establish weighted elliptic estimates that allow us to find initial data in some weighted Sobolev spaces with weight $w(x)=(1+|x|^2)^{\mu}, \mu \geq 2$, and we show this propagates within short time; in other words, we are able to prove weighted energy estimates for compressible water waves. These results serve as good preparation for proving long time existence also for compressible water waves.