Albert Milani

Linear and Quasi-linear Evolution Equations in Hilbert Spaces

This book considers evolution equations of hyperbolic and parabolic type. These equations are studied from a common point of view, using elementary methods, such as that of energy estimates, which prove to be quite versatile. The authors emphasize the Cauchy problem and present a unified theory for the treatment of these equations. In particular, they provide local and global existence results, as well as strong well-posedness and asymptotic behavior results for the Cauchy problem for quasi-linear equations. Solutions of linear equations are constructed explicitly, using the Galerkin method; the linear theory is then applied to quasi-linear equations, by means of a linearization and fixed-point technique. The authors also compare hyperbolic and parabolic problems, both in terms of singular perturbations, on compact time intervals, and asymptotically, in terms of the diffusion phenomenon, with new results on decay estimates for strong solutions of homogeneous quasi-linear equations of each type. This textbook presents a valuable introduction to topics in the theory of evolution equations, suitable for advanced graduate students. The exposition is largely self-contained. The initial chapter reviews the essential material from functional analysis. New ideas are introduced along with their context. Proofs are detailed and carefully presented. The book concludes with a chapter on applications of the theory to Maxwells equations and von Karmans equations.

Evolution Equations of von Karman Type

Operators and Spaces.- Weak Solutions.- Strong Solutions, m + k _ 4.- Semi-Strong Solutions, m = 2, k = 1.

Semi-strong Solutions, m = 2, k = 1

In this chapter we prove Theorem 1.4.3 on the existence and uniqueness of semi-strong solutions of problem (VKH) when m = 2 (recall that, by Definition 1.4.1, if m = 2 there is only one kind of semi-strong solution, corresponding to k = 1). Accordingly, we assume that

The Hardy Space \(\mathcal{H}^{1}\) and the Case m = 1

\displaystyle{ u_{0} \in H^{3}\,,\qquad u_{ 1} \in H^{1}\,,\qquad \varphi \in S_{ 2,1}(T) = C([0,T];H^{5}) }

The Parabolic Case

(4.1) [recall ( 1.137)], and look for solutions of problem (VKH) in the space \(\mathcal{X}_{2,1}(\tau )\), for some τ ∈ ]0, T].

Exponential attractors and inertial manifolds for singular perturbations of the Cahn–Hilliard equations

In this chapter we introduce the function spaces in which we build our solution theory for problems (VKH) and (VKP), and study the main properties of the operator N defined in (8) in these spaces.

Long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term—III

In this chapter we first review a number of results on the regularity of the functions N = N(u1, … , u m ) and f = f(u) in the framework of the Hardy space \(\mathcal{H}^{1}\), and then use these results to prove the well-posedness of the von Karman equations (3) and (4) in \(\mathbb{R}^{2}\).

A Remark on the Sobolev Regularity of Classical Solutions of Strongly Elliptic Equations

In this chapter we assume that m ≥ 2, k ≥ 1, with m + k ≥ 4, and prove Theorem 1.4.2 on the uniformly local strong well-posedness of problem (VKH) in the space \(\mathcal{X}_{m,k}(\tau )\), for some τ ∈ ]0, T] independent of k.