On the Cauchy problem for some hyperbolic operator with double characteristics

2006 ◽  
pp. 29-44 ◽  
Author(s):  
Enrico Bernardi ◽  
Antonio Bove
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Operator ◽  
The Cauchy Problem

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 fundamental solution of the Cauchy problem for a hyperbolic operator

2012 ◽  
Author(s):  
Ж.Д. Тотиева
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Hyperbolic Operator ◽  
The Cauchy Problem

В работе исследуется фундаментальное решение обобщенной задачи Коши для одного гиперболического оператора, и изучены некоторые свойства фундаментального решения.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

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