Finding a solutions for an equation

ap.evaluation of pdes – Hipoellipticity or parabolic regularity for vector bundles Answer

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ap.evaluation of pdes – Hipoellipticity or parabolic regularity for vector bundles

Let $E to M$ breathe a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not essentially compact). Let $H : Gamma(E) to Gamma(E)$ breathe a differential operator with flush coefficients such that its principal attribute is $g$, the Riemann tensor of $M$ (so $H$ is a generalized Laplacian). If $p : (0, infty) instances M to M$ is the unaffected projection, let $F = p^* E$ breathe the pull-back bundle. Let $u$ breathe a piece in $F$ with distributional values such that $(partial_t – H) u = 0$ within the distributional sense.

Are there ready-made instruments out there that will enable me to resolve that $u$ is flush on $(0, infty) instances M$?

The query reminds one of many ideas of hypoellipticity or parabolic regularity for capabilities. I might attempt to mimick these and bear analogous outcomes for sections, however for positive I’m not the primary one to necessity them, so possibly they’ve already been obtained and I simply do not know the place to search for them (a Google search did not ameliorate both). I too consider that this drawback is not only some brisk corollary of the same outcomes for capabilities.

(In my concrete drawback $u$ is a few $L^2 _{loc}$-integrable part. $(0, infty)$ might very properly breathe changed with $mathbb R$, of passage, however in my labor I necessity the warmth semigroup afterward.)

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