## Approximation of Nonlinear Evolution SystemsApproximation of Nonlinear Evolution Systems |

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analysis applied approximation assumed Banach space boundary bounded Brézis Cauchy problem Chapter compact condition continuous function continuously embedded convergence convex Corollary deﬁned Deﬁnition denote dense derivative differential equations Dirichlet domain elliptic embedded equivalent estimates evolution operators existence ﬁeld ﬁnal ﬁnite ﬁnite-dimensional ﬁnite-element ﬁrst ﬂuid follows given hence Hilbert space holds inequality initial datum integration isomorphism Lebesgue integrable Lemma Lipschitz continuous lower semicontinuity mapping Math maximal monotone method models modiﬁcations Moreover N-width Navier—Stokes nonlinear nonnegative normed linear space Note obtain partial differential equations partitions pointwise polynomial porous-medium equation Proof Proposition pseudomonotone quasi-stability regularity relation respectively result right-hand side satisﬁes satisfying Section self-adjoint semidiscrete semigroup sequence smoothing Sobolev Sobolev spaces speciﬁed stability Stefan problem subset subspace Suppose tensor term Theorem theory two-phase Stefan problem unique variables veriﬁcation weak solution weakly