optimality of energy estimates for non smooth metric - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:08:12Z http://mathoverflow.net/feeds/question/73676 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73676/optimality-of-energy-estimates-for-non-smooth-metric optimality of energy estimates for non smooth metric michael 2011-08-25T17:30:13Z 2011-08-28T12:57:37Z <p>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).</p> http://mathoverflow.net/questions/73676/optimality-of-energy-estimates-for-non-smooth-metric/73683#73683 Answer by michael for optimality of energy estimates for non smooth metric michael 2011-08-25T18:15:04Z 2011-08-25T20:58:45Z <p>-> question reedited</p>