Evolutionary problems in non-cylindrical domains. (English) Zbl 1473.35342
Vespri, Vincenzo (ed.) et al., Harnack inequalities and nonlinear operators. Proceedings of the INdAM conference to celebrate the 70th birthday of Emmanuele DiBenedetto. Cham: Springer. Springer INdAM Ser. 46, 43-60 (2021).
Summary: This survey article presents an existence theory developed in [the authors with T. Singer, SIAM J. Math. Anal. 50, No. 3, 3007–3057 (2018; Zbl 1390.35125)] for vector-valued gradient flows of integral functionals in bounded non-cylindrical domains \(E\subset\mathbb{R}^n\times [0,T)\). The associated system of differential equations takes the form
\[ \partial_t u - \operatorname{div} D_\xi f(x,u,Du) = -D_u f(x,u,Du) \quad \text{in }E, \]
for an integrand \(f(x, u, Du)\) that is convex and coercive with respect to the \(W^{1, p}\)-norm for \(p > 1\).
For the entire collection see [Zbl 1470.35004].
For the entire collection see [Zbl 1470.35004].
MSC:
| 35K59 | Quasilinear parabolic equations |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35B45 | A priori estimates in context of PDEs |
Citations:
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