Partial Differential Equations as Equations in Infinite Dimensions
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The study of infinite–dimensional equations is motivated by applications to partial differential equations. We describe two methods used for deterministic equations. One involves the study of mild solutions with the help of a semigroup generated by the unbounded operator on a Banach space, as in the work of Pazy (Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York 1983). We illustrate this approach in an example of the heat equation cast as an abstract Cauchy problem, where the state space of the solution is an infinite-dimensional Banach space, and the coefficient, even in this linear case, is an unbounded operator. We present here a comprehensive review of the properties of semigroups of operators. In the other method, the unbounded operator is recast as a bounded operator from its domain Hilbert space V to its range V ∗, the continuous dual of V, where the equation is defined. However, the initial condition is in a Hilbert space H, with V↪H↪V ∗, where the embeddings are dense and continuous. This approach requires a condition of coercivity to bring the solution at time t>0 into H, the space of the initial condition. We present here the approach of Lions (Équations Différentielles Opérationelles et Problèmes aux Limites. Springer, Berlin, 1961), to show that the solution is a continuous H-valued function on [0,T]. In the presence of non-linear coefficients, the solution may not exist due to the failure of the Peano theorem (since the Arzela–Ascoli theorem is invalid in infinite dimensions), and to secure the existence result we will assume in later chapters compactness of the embeddings or of the semigroup of operators.
KeywordsHilbert Space Banach Space Partial Differential Equation Linear Operator Heat Equation
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