Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on $$\mathbb {R}^n$$ R n

Ascanelli, Alessia; Coriasco, Sandro

doi:10.1007/s11868-015-0132-x

Cited by 6 publications

(51 citation statements)

References 25 publications

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“…For the convenience of the reader, here we recall a few results from [3]. Let us consider a sequence tϕ j u jPN of regular SG phase functions ϕ j px, ξq P P r pτ j q, j P N, with…”

Section: Multi-products Of Regular Sg Phase Functions and Of Regular mentioning

confidence: 99%

“…As a simple realization of a sequence of phase functions satisfying (2.10) and (2.13), we recall the following example, see [3] and [25]. Example 2.18.…”

Section: Eikonal Equations and Hamilton-jacobi Systems In Sg Classesmentioning

confidence: 99%

“…The following result about existence and uniqueness of a solution Upt, sq to the Cauchy problem (4.1) is a SG variant of the classical Duhamel formula, see [3,10,12]. Proposition 4.1.…”

Section: Cauchy Problems For Weakly Sg-hyperbolic Systemsmentioning

confidence: 99%

“…Moreover, we state several fundamental theorems for SG Fourier integral operators. Further details can be found, e.g., in [3,10,23,25,36] and the references therein, from which we took most of the materials included in this section. Here and in what follows, A -B means that A B and B A, where A B means that A ď c¨B, for a suitable constant c ą 0.…”

Section: Introductionmentioning

confidence: 99%

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Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on $$\mathbb {R}^n$$

2020

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We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R n . We prove wellposedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To this aim, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such expressions. 2010 Mathematics Subject Classification. Primary: 58J40; Secondary: 35S05, 35S30, 47G30, 58J45. Key words and phrases. Fourier integral operator, multi-product, hyperbolic Cauchy problem, involutive characteristics, stochastic PDEs. of operators such that $ & % LEpt, sq " 0 0 ď s ď t ď T, Eps, sq " I, s P r0, Tq, so obtaining by Duhamel's formula the solution Uptq " Ept, 0qG`i ż t 0Ept, sqFpsqdsto the system and then the solution uptq to the corresponding higher order Cauchy problem (1.1) by the reduction procedure.In the present paper we consider the Cauchy problem (1.1) in the SG setting, that is, under the growth condition (1.3), the hyperbolicity condition (1.4) and the involutiveness assumption (1.5) for suitable parameter-dependent, real-valued symbols b jk , d jk P C 8 pr0, Ts; S 0,0 pR 2n qq, j, k P N. The involutiveness of the characteristic roots of the operator, or, equivalently, of the eigenvalues of the (diagonal) principal part of the corresponding first order system is here the main assumption (see Assumption I in Section 3.3 for the precise statement of such condition).We obtain several results concerning the Cauchy problem (1.1): ‚ we prove, in Theorem 5.8, the existence, for every f P C 8 pr0, Ts; H r,ρ pR n qq and g k P H r`m´1´k,ρ`m´1´k pR n q, k " 0, . . . , m´1, of a uniquefor a suitably small 0 ă T 1 ď T, solution of (1.1); we also provide, in (5.15), the explicit expression of u, using SG Fourier integral operators (SG FIOs); ‚ we give a precise description, in Theorem 5.21, of a global wave-front set of the solution, under a (mild) additional condition on the operator L; namely, we prove that, when L is SG-classical, the (smoothness and decay) singularities of the solution of (1.1) with f " 0 are points of unions of arcs of bicharacteristics, generated by the phase functions of the SG FIOs appearing in the expression (5.15), and emanating from (smoothness and decay) singularities of the Cauchy data g k , k " 0, . . . , m´1; ‚ we deal, in Theorem 6.3, with a stochastic version of the Cauchy proble...

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“…For the convenience of the reader, here we recall a few results from [3]. Let us consider a sequence tϕ j u jPN of regular SG phase functions ϕ j px, ξq P P r pτ j q, j P N, with…”

Section: Multi-products Of Regular Sg Phase Functions and Of Regular mentioning

confidence: 99%

“…As a simple realization of a sequence of phase functions satisfying (2.10) and (2.13), we recall the following example, see [3] and [25]. Example 2.18.…”

Section: Eikonal Equations and Hamilton-jacobi Systems In Sg Classesmentioning

confidence: 99%

“…The following result about existence and uniqueness of a solution Upt, sq to the Cauchy problem (4.1) is a SG variant of the classical Duhamel formula, see [3,10,12]. Proposition 4.1.…”

Section: Cauchy Problems For Weakly Sg-hyperbolic Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on $$\mathbb {R}^n$$

2020

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“…To construct the fundamental solution of (1.1) we will need, on one hand, to perform compositions between pseudo-differential operators and Fourier integral operators of SG type, using the theory developed in [13], and, on the other hand, compositions between Fourier integral operators of SG type with possibly different phase functions. The latter can be achieved using the composition results recently obtained in [5], with the aim of applying them in the present paper. The paper [5] is quite technical, so here we will only recall and make use of the main composition theorems coming from the theory developed there.…”

Section: Introductionmentioning

confidence: 99%

Solution theory to semilinear hyperbolic stochastic partial differential equations with polynomially bounded coefficients

Ascanelli

Coriasco

Süß³

2019

Nonlinear Analysis

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We study mild solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable hyperbolicity hypotheses on the linear part. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that mild solutions exist and are unique in suitably chosen functional classes. More precisely, function-valued solutions are obtained, as well as a regularity result. 1 2 ALESSIA ASCANELLI, SANDRO CORIASCO, AND ANDRÉ SÜSS uniformly bounded coefficients depending on x ∈ R d , d ≥ 1. There, sufficient conditions on the stochastic terṁ Ξ and on the coefficients of A are given, in order to find a unique function-valued solution. In the present paper we show existence and uniqueness of a function-valued solution to a wider class of semilinear hyperbolic SPDEs, with possibly unbounded coefficients depending on (t,A disadvantage of the function(al-spaces)-valued solution u of (1.1) sketched above is that, in general, it cannot be evaluated in the spatial argument (usually, it is a random element in the t (that is, time) parameter, taking values in a L p (R d )-modeled Hilbert or Banach space). Then, an alternative approach focuses instead on the concept of stochastic integral with respect to a martingale measure. That is, the stochastic integral in (1.2) is defined through the martingale measure derived from the random noiseΞ. Here one obtains a so-called random-field solution, that is, u is defined as a map associating a random variable to each (t, x) ∈ [0, T 0 ]×R d , where T 0 > 0 is the time horizon of the equation. For more details, see, e.g., the classical references [10,17,34], and the recent papers [2,7], where the existence of a random-field solution in the case of linear hyperbolic SPDEs with (t, x)-dependent coefficients has been shown. On the other hand, a disadvantage of such random-field solution u of (1.1) is that its construction for non-linear equations is based on the stationarity condition Λ = Λ(t − s, x − y), which is fulfilled by SPDEs with constant coefficients, but cannot be assumed if we want to deal with more general linear operators L in (1.1), indeed admitting variable coefficients. Namely, we have shown in [2] how random-field solutions can be constructed for arbitrary order, linear (weakly) hyperbolic SPDEs, with (possibly unbounded) coefficients smoothly depending onIt is remarkable that, in many cases, the theory of integration with respect to processes taking values in functional spaces, and the theory of integration with respect to martingale measures, lead to the same solution u (in some sense) of an SPDE, see [18] for a precise comparison. We conclude the paper showing a result of that kind, comparing the function-valued solutions to (1.1) obtained in the present paper, in the special case of the linear equations, with the random-field solutions of the same equation found in [2].We remark that in the present paper, as well as in [2,7], the...

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Weyl Law on Asymptotically Euclidean Manifolds

Coriasco

Doll

2020

Ann. Henri Poincaré

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We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator $$Q=(1+|x|^2)(1-\varDelta )$$ Q = ( 1 + | x | 2 ) ( 1 - Δ ) on $$\mathbb {R}^d$$ R d .

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Fourier integral operators algebra and fundamental solutions to hyperbolic systems with polynomially bounded coefficients on $$\mathbb {R}^n$$ R n

Cited by 6 publications

References 25 publications

Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on $$\mathbb {R}^n$$

Deterministic and stochastic Cauchy problems for a class of weakly hyperbolic operators on $$\mathbb {R}^n$$

Solution theory to semilinear hyperbolic stochastic partial differential equations with polynomially bounded coefficients

Weyl Law on Asymptotically Euclidean Manifolds

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