On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity
On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity
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高阶正则性 Levi 条件下二阶双曲方程的能量估计
DOI：
10.1007/s11565-011-0121-9
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Tang Bao Ngoc Bui
中科院分区：
文献类型：
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作者：
Fumihiko Hirosawa;Tang Bao Ngoc Bui
考虑系数依赖于时间的二阶齐次双曲型方程的能量估计。能量守恒的性质在常系数的情况下成立，但在变系数的情况下一般不成立;事实上，在这种情况下，能量可以是无界的ast→ ∞。对于波型方程，精确地研究了广义能量守恒（GEC）系数的条件，即只有传播速度是可变的，广义能量守恒（GEC）是能量关于时间均匀等价的。然而，对于一般的齐次双曲型方程，并不存在关于系数的相同条件（GEC）。本文的主要目的是给出提供（GEC）的系数的附加条件，由于它们与弱双曲型方程适定性的通常Levi条件的意义基本相同，所以称之为Ck型Levi条件.
We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded ast→ ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called asCk-type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.