Somewhat surprisingly , given their wide range of practical uses , mathematicians have not yet proven that in three dimensions solutions always exist ( existence ) , or that if they do exist , then they do not contain any singularity ( or infinity or discontinuity ) ( smoothness ) . Somewhat surprisingly , given their wide range of practical uses , mathematicians have not yet proven that in three dimensions solutions always exist ( existence ) , or that if they do exist , then they do not contain any singularity ( smoothness ) .