Parabolic systems revisited

Pascal Auscher, Moritz Egert

Published 2022-11-30Version 1

We propose a universal construction of Green operators for parabolic systems with rough, unbounded coefficients via a variational formulation when time describes the real line. Lower order coefficients are controlled in mixed time-space Lebesgue norms with the critical (homogeneous) or subcritical (inhomogeneous) exponents. Well-posedness of the Cauchy problem comes as a consequence along with representations by fundamental solution operators. We prove L^2 off-diagonal estimates for these fundamental solution operators, which is new under critical assumptions on lower order coefficients even when the coefficients are real-valued, and obtain pointwise Gaussian upper bounds under local bounds for weak solutions, recovering in particular Aronson's upper estimates. The scheme is general enough to allow systems with higher order elliptic parts on full space or second order elliptic parts on Sobolev spaces with boundary conditions. Another new feature is that the control on coefficients can be relaxed to mixed Lorentz norms, even though this choice may lead to counterexamples for regularity of weak solutions.