LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

2014 ◽  
Vol 11 (01) ◽  
pp. 185-213 ◽  
Author(s):  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Energy Estimates ◽  
Characteristic Set ◽  
Finite Speed ◽  
Hyperbolic Operators ◽  
New Energy ◽  
The Cauchy Problem ◽  
Well Posed

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

scholarly journals Cauchy problem for effectively hyperbolic operators with triple characteristics of variable multiplicity

2015 ◽  
Vol 12 (03) ◽  
pp. 535-579 ◽  
Author(s):  
Enrico Bernardi ◽  
Antonio Bove ◽  
Vesselin Petkov
Keyword(s):  
Cauchy Problem ◽  
Principal Symbol ◽  
Energy Estimates ◽  
Hyperbolic Operator ◽  
Third Order ◽  
Hyperbolic Operators ◽  
The Cauchy Problem ◽  
Loss Of Regularity ◽  
Well Posed ◽  
Lower Order Terms

We study a class of third-order hyperbolic operators P in G = {(t, x): 0 ≤ t ≤ T, x ∈ U ⋐ ℝn} with triple characteristics at ρ = (0, x0, ξ), ξ ∈ ℝn ∖{0}. We consider the case when the fundamental matrix of the principal symbol of P at ρ has a couple of non-vanishing real eigenvalues. Such operators are called effectively hyperbolic. Ivrii introduced the conjecture that every effectively hyperbolic operator is strongly hyperbolic, that is the Cauchy problem for P + Q is locally well posed for any lower order terms Q. This conjecture has been solved for operators having at most double characteristics and for operators with triple characteristics in the case when the principal symbol admits a factorization. A strongly hyperbolic operator in G could have triple characteristics in G only for t = 0 or for t = T. We prove that the operators in our class are strongly hyperbolic if T is small enough. Our proof is based on energy estimates with a loss of regularity.

scholarly journals On the Cauchy problem for Dt2 − Dx(b(t)a(x))Dx

2020 ◽  
Vol 17 (01) ◽  
pp. 75-122
Author(s):  
Ferruccio Colombini ◽  
Tatsuo Nishitani
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Second Order ◽  
Gevrey Class ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Two Independent Variables

We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].

scholarly journals Inhomogeneous Gevrey ultradistributions and Cauchy problem

2006 ◽  
Vol 133 (31) ◽  
pp. 176-186
Author(s):  
Daniela Calvo ◽  
L. Rodino
Keyword(s):  
Cauchy Problem ◽  
Weight Function ◽  
Differential Operators ◽  
Subject Classification ◽  
Well Posedness ◽  
Gevrey Classes ◽  
Partial Differential Operators ◽  
Hyperbolic Operators ◽  
The Cauchy Problem ◽  
Constant Coefficients

After a short survey on Gevrey functions and ultradistributions, we present the inhomogeneous Gevrey ultradistributions introduced recently by the authors in collaboration with A. Morando, cf. [7]. Their definition depends on a given weight function ?, satisfying suitable hypotheses, according to Liess-Rodino [16]. As an application, we define (s, ?)-hyperbolic partial differential operators with constant coefficients (for s > 1), and prove for them the well-posedness of the Cauchy problem in the frame of the corresponding inhomogeneous ultradistributions. This sets in the dual spaces a similar result of Calvo [4] in the inhomogeneous Gevrey classes, that in turn extends a previous result of Larsson [14] for weakly hyperbolic operators in standard homogeneous Gevrey classes. AMS Mathematics Subject Classification (2000): 46F05, 35E15, 35S05.

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

1990 ◽  
Vol 427 (1872) ◽  
pp. 221-239 ◽  
Keyword(s):  
Cauchy Problem ◽  
Perfect Fluid ◽  
Initial Value Problem ◽  
Einstein Equations ◽  
Initial Value ◽  
Fluid Source ◽  
Null Hypersurfaces ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Characteristic Initial Value Problem

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

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