We prove L2-maximal regularity of the linear non-autonomous evolutionary Cauchy problem \dot u(t) + A(t)u(t) = f(t){\text{ for a}}{\text{.e}}{\text{. }}t \in \left[ {0,T} \right],{\text{ }}u(0) = {u_0}, where the operator A(t) arises from a time depending sesquilinear form a(t, ·, ·) on a Hilbert space H with constant domain V. We prove the maximal regularity in H when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of H., Ahmed Sani, Hafida Laasri., and Obsahuje seznam literatury
Several abstract model problems of elliptic and parabolic type with inhomogeneous initial and boundary data are discussed. By means of a variant of the Dore-Venni theorem, real and complex interpolation, and trace theorems, optimal Lp-regularity is shown. By means of this purely operator theoretic approach, classical results on Lp-regularity of the diffusion equation with inhomogeneous Dirichlet or Neumann or Robin condition are recovered. An application to a dynamic boundary value problem with surface diffusion for the diffusion equation is included.
Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and $L^2$-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented in the special situation only.
We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem $$ ({\rm P}) \begin {cases} \dot {u}(t)+A(t)u(t)=f(t)\quad t\text {-a.e. on} [0,\tau ], u(0)=0, \end {cases} $$ where $A\colon [0,\tau ]\to \mathcal {L}(X,D)$ is a bounded and strongly measurable function and $X$, $D$ are Banach spaces such that $D\underset {d}\to {\hookrightarrow }X$. Our main concern is to characterize $L^p$-maximal regularity and to give an explicit approximation of the problem (P).