You are here

Abstract Cauchy Problems: Three Approaches by Irina V. Melnikova, Alexei Filinkov PDF

By Irina V. Melnikova, Alexei Filinkov

ISBN-10: 1584882506

ISBN-13: 9781584882503

Proper to various mathematical versions in physics, engineering, and finance, this quantity reviews Cauchy difficulties that aren't well-posed within the classical feel. It brings jointly and examines 3 significant techniques to treating such difficulties: semigroup equipment, summary distribution equipment, and regularization tools. even though broadly built during the last decade, the authors supply a different, self-contained account of those tools and display the profound connections among them. available to starting graduate scholars, this quantity brings jointly many various principles to function a reference on sleek equipment for summary linear evolution equations.

Show description

Read Online or Download Abstract Cauchy Problems: Three Approaches PDF

Best functional analysis books

Jussi Behrndt, Karl-Heinz Förster, Heinz Langer, Carsten's Spectral theory in inner product spaces and applications: PDF

This booklet incorporates a choice of contemporary study papers originating from the sixth Workshop on Operator conception in Krein areas and Operator Polynomials, which used to be held on the TU Berlin, Germany, December 14 to 17, 2006. The contributions during this quantity are dedicated to spectral and perturbation idea of linear operators in areas with an internal product, generalized Nevanlinna services and difficulties and purposes within the box of differential equations.

Get Methods for Solving Inverse Problems in Mathematical Physics PDF

Constructing an method of the query of lifestyles, forte and balance of recommendations, this paintings provides a scientific elaboration of the speculation of inverse difficulties for all primary forms of partial differential equations. It covers up to date tools of linear and nonlinear research, the speculation of differential equations in Banach areas, functions of sensible research, and semigroup idea.

Download e-book for iPad: Born-Jordan Quantization: Theory and Applications by Maurice A. de Gosson

This e-book offers a finished mathematical examine of the operators at the back of the Born–Jordan quantization scheme. The Schrödinger and Heisenberg photographs of quantum mechanics are an identical provided that the Born–Jordan scheme is used. therefore, Born–Jordan quantization offers the one bodily constant quantization scheme, in place of the Weyl quantization wide-spread by means of physicists.

Read e-book online Spaces of Continuous Functions PDF

The gap C(X) of all non-stop services on a compact house X consists of the constitution of a normed vector house, an algebra and a lattice. at the one hand we research the family among those buildings and the topology of X, nevertheless we talk about a few classical effects in response to which an algebra or a vector lattice will be represented as a C(X).

Extra info for Abstract Cauchy Problems: Three Approaches

Example text

An x . 2 Let n ∈ N. The Cauchy problem (CP) is said to be (n, ω)-well-posed on E if for any x ∈ E ⊆ D(An+1 ) ©2001 CRC Press LLC ©2001 CRC Press LLC (a) there exists a unique solution u(·) ∈ C [0, ∞], D(A) ∩ C 1 [0, ∞], X ; (b) ∃K > 0, ω ∈ R : u(t) ≤ Keωt x An . If E = D(An+1 ), then we say that the problem (CP) is (n, ω)-well-posed. 4 Let A be a densely defined linear operator on X with nonempty resolvent set. Then the following statements are equivalent: (I) A is the generator of an n-times integrated semigroup {V (t), t ≥ 0}; (II) the Cauchy problem (CP) is (n, ω)-well-posed.

13) is satisfied as described in Cases 1–3 according to the choice of the initial data. 21) where u(t) u (t) w(t) = Φ= 0 I A 0 ∈ L2 (Ω) × L2 (Ω), D(Φ) = D(A) × L2 (Ω). 21) can be formally written as w(t) = U (t) = ©2001 CRC Press LLC ©2001 CRC Press LLC u0 u1 := C(t) C (t) C(t)u0 + S(t)u1 C (t)u0 + S (t)u1 , S(t) S (t) u0 u1 t ≥ 0, u0 , u1 ∈ L2 (Ω), and noting that for any v ∈ L2 (Ω), S (t)v = C(t)v, we write U in the form C(t) S(t) C (t) C(t) U (t) = t ≥ 0. , From the definition of C it is clear that C is not necessarily differentiable in t on L2 (Ω), implying that the operators U (t) are in general unbounded on L2 (Ω) × L2 (Ω), and therefore they do not form a C0 -semigroup on this space.

3) To prove that D(A) = X, we consider the set b U := U (τ )udτ, x ∈ X, b > a > 0 . va,b = a We show that U ⊂ D(A): h−1 U (h) − I va,b = b h−1 [U (h + τ ) − U (τ )] xdτ a = b+h h−1 b U (t)xdt − a+h = b+h h−1 ©2001 CRC Press LLC ©2001 CRC Press LLC a U (τ )xdτ − b → U (τ )xdτ a U (τ )xdτ a+h U (b) − U (a) x as h → 0. Now we will show that U = X. Suppose that there exists f ∈ X ∗ such that f (U) = 0, f ≡ 0. Then b ∀b > a > 0, f (va,b ) = f (U (τ )x)dτ = 0. a Hence ∀x ∈ X, f (U (τ )x) = 0, τ > 0, and f (U (τ )x) →τ →0 f (x) = 0, that is f ≡ 0.

Download PDF sample

Abstract Cauchy Problems: Three Approaches by Irina V. Melnikova, Alexei Filinkov

by Jeff

Rated 4.18 of 5 – based on 48 votes