Consider the linear (geometric) wave equation in dimension (3+1) with non smooth background metric $g$ say $g \in L^\infty_t H^3_x$ and $\partial_t g \in L^\infty H^2_x$, then energy estimates enable to propagate sobolev regularity of the initial data set till $H^3\times H^2$, is this sharp in the following sense : can you find a metric with this regularity and an initial data set $u_0, u_1$ in say $H^4 \times H^3$ s.t. the corresponding solution of the linear wave equation does not lie even locally in $L^\infty_t H^4_x$ but only in $L^\infty_t H^3_x$ ? Seems to me true but i fail to construct a counterexample. (I would like to apologize for not understanding how to reedit from another computer and a unregistered account, sorry).
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$\begingroup$ I don't think your scaling argument is correct. The fundamental energy estimate for the linear wave equation $g^{ij}\partial^2_{ij}\phi = 0$ gives that $$ E[\phi](t) - E[\phi](0) \leq \int |\partial g| |\partial \phi|^2 dx dt$$ so the more natural regularity assumption on the metric should be $\partial g\in L^1_tL^\infty_x$. Just $g\in L^\infty_tH^3_x$ does not give you any energy estimate, since in the energy estimate you need to take a time derivative of the metric. If you use the $L^2$ Sobolev scale, you also have to specify the number of spatial dimensions. $\endgroup$– Willie WongAug 25, 2011 at 18:05
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-> question reedited
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$\begingroup$ Actually, for spatial dimension $n$, the general energy estimate is $$\|(u, \partial_tu)\|_{H^{k+1}\times H^k} \lesssim E_0 \exp \int_0^t \|\partial g\|_{\infty} + \|g\|_{W^{k,p}} ds $$ where $p = (k-1)/n$ if $n \geq 2(k-1)$ or $p = 2$ otherwise if you use a Moser type product estimate. (BTW, this is what gives the LWP for quasilinear wave equations: if $g = g(u)$ you get LWP in $H^k$ for $k > n/2 + 1$ and if $g = g(u,\partial u)$ you need $n/2 + 2$.) The "covariant" doesn't matter at all since it just adds a term $\partial g \partial u$ to the RHS. $\endgroup$ Aug 25, 2011 at 19:25