Linear and Quasi-linear Evolution Equations in Hilbert Spaces

Cherrier, Pascal; Milani, Albert

doi:10.1090/gsm/135

Cited by 36 publications

(25 citation statements)

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“…It is an open problem, however, to establish if the last claim of Theorem 2.6 (passage to the limit in the equation) still applies when p = 2. A positive answer would settle the long-standing open question of global existence of weak solutions for this kind of equations (see [1]).…”

Section: Sinementioning

confidence: 99%

“…In this paper we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic PDEs having the formal structure (1) w ′′ (t, x) = −∇W w(t, ·) (x), (t, x) ∈ R + × R n , with prescribed initial conditions (2) w(0, x) = w 0 (x), w ′ (0, x) = w 1 (x).…”

Section: Introductionmentioning

confidence: 99%

“…While a precise setting with all formal details and our main results are given in Section 2, here we confine ourselves to a rather informal description of our approach, focusing on the main ideas that lie behind it and on the possible new perspectives that it opens up, especially some new connections between the variational world and hyperbolic PDEs of the kind (1). In (1), ∇W is the Gâteaux derivative of a functional (e.g. one from the Calculus of Variations) W : W → [0, ∞), where W is some Banach space of functions in R n , typically a Sobolev space.…”

Section: Introductionmentioning

confidence: 99%

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A minimization approach to hyperbolic Cauchy problems

Serra¹,

Tilli²

2016

J. Eur. Math. Soc.

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Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of functions that minimize suitable functionals in space-time (where the initial data of the Cauchy Problem serve as prescribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the Calculus of Variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that fit in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.Mathematics subject classification: 35L70, 35L90, 35L15, 49J45.

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Section: Sinementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A minimization approach to hyperbolic Cauchy problems

Serra¹,

Tilli²

2016

J. Eur. Math. Soc.

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“…On the other hand, we note that (as in [21]) Theorem 2.5 holds under the sole assumption f ∈ L 2 loc ([0, ∞); L 2 ) on the source term (which is the natural assumption if one seeks solutions of (1) with finite energy -see e.g. [3,10,21]).…”

Section: Functional Setting and Main Resultsmentioning

confidence: 97%

An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

Tentarelli

Tilli

2019

Journal of Differential Equations

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We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a first-order-in-time linear term. The paper carries on the research program initiated in [14], and developed in [15,21], on the De Giorgi approach to hyperbolic equations.Hence, the link with (1) is immediate: as ε ↓ 0, supposing that f ε → f and w ε → w, one formally obtains (1) (and (2)) in the limit.A first comment is in order. The paper provides a purely variational method for proving existence of (weak) solutions to second order hyperbolic PDEs with dissipation, that is, wave-like

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“…The existence of global weak solutions to problem (VKH) can be established by a straightforward generalization of J. L. Lions' result mentioned earlier for the case m = 1; likewise, the existence and uniqueness of local semi-strong solutions can be established by methods similar to those we used for strong and regular solutions in chapter 7, sct. 2, of our book [5]. We claim:…”

mentioning

confidence: 86%