The regularity of general parabolic systems with degenerate diffusion

Higher integrability for parabolic systems with non-standard growth and degenerate diffusions

“…a structure prescribing that the gradient non-linearities depend only on the explicit scalar quantity |Du|, or a related partial regularity theory for parabolic systems with degenerate p(z)-diffusion (cf. [14], [10] for related results in the case p ≡ const).…”

Very weak solutions of subquadratic parabolic systems with non-standardp(x,t)-growth

“…To this aim we fix a cylinder Q (6) . In the following, we will write B (k) for the projection of Q (k) in x direction and Λ (k) for the projection of Q (k) in t direction for k ∈ {0, · · · , 6}.…”

“…In the following, we will write B (k) for the projection of Q (k) in x direction and Λ (k) for the projection of Q (k) in t direction for k ∈ {0, · · · , 6}. Denoting by p 1 and p 2 the minimum and maximum of p(·) over the cylinder Q (6) , i.e. (6) p(·) and p 2 = sup Q (6) p(·), and taking into account that p(·) ≤ 2 and that ω is concave, we find that…”

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

“…It is well-known that if |F | ∈ L p (Ω T ), then the equation (1) has a unique weak solution u satisfying the following Calderón-Zygmund estimate (5) sup…”

“…For instance this was done in [3] for nonlinear elliptic equations which made use of estimates below the natural growth exponent and Hodge decompositions. In the parabolic case such tools can be replaced by the methods in [5] and might be done in future work.…”