scholarly journals Cauchy problem for hyperbolic differential operators with double characteristic roots

1980 ◽  
Vol 56 (2) ◽  
pp. 67-69
Author(s):  
Katsuju Igari
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Characteristic Roots ◽  
Double Characteristic

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.

Locally excited magnethohydrodynamic waves in a uniform cylindrical plasma waveguide

1977 ◽  
Vol 17 (2) ◽  
pp. 281-300 ◽  
Author(s):  
W. N-C. Sy
Keyword(s):  
Differential Operators ◽  
Current Source ◽  
Plasma Waveguide ◽  
Mhd Waves ◽  
Cylindrical Plasma ◽  
Characteristic Roots ◽  
Guided Modes ◽  
Homogeneous Wave ◽  
Helmholtz Operators ◽  
Homogeneous Wave Equation

The guided modes of Woods' magnetohydrodynamic waveguide for a uniform, cylindrical plasma are shown to satisfy a homogeneous wave equation whose differential operator is the product of eight Helmholtz operators. The propagation constants of the Helmholtz operators are the characteristic roots of an 8 × 8 matrix which is derived and written down explicitly. This reformulated theory is extended to include localized sources which excite the guided modes. For certain cases, the Green's functions for the differential operators can be represented by Dini expansions in terms of modal eigenfunctions, which manifestly satisfy the boundary conditions. For the case of MHD waves excited by an azimuthally symmetric current source in a resistive, pressureless, inviscid, fully ionized plasma, a detailed solution is obtained which is in good qualitative agreement with experiments.

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].

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