Very weak solutions to hypoelliptic wave equations
Very weak solutions to hypoelliptic wave equations
复制标题
亚椭圆波动方程的极弱解
DOI：
10.1016/j.jde.2019.09.020
复制
发表时间：
2018
影响因子：
2.4
通讯作者：
Nurgissa Yessirkegenov
中科院分区：
文献类型：
--
作者：
Michael Ruzhansky;Nurgissa Yessirkegenov
本文研究了分次李群上亚椭圆齐次左不变算子波动方程的Cauchy问题，其中非负传播速度是正则的、Hölder的和分布的.对于Hölder系数，我们得到了分次群上与罗克兰算子相关的超分布空间的适定性.在传播速度是一个分布的情况下，我们采用柯西问题的“非常弱解”的概念，这已经在[12]和[20]中成功地用于类似的情况。我们证明了具有分布系数的波动方程的Cauchy问题在适当的意义下存在唯一的“很弱解”，当经典解或分布解存在时，该解与经典解或分布解一致.例子包括随时间变化的波动方程的子拉普拉斯海森堡组或一般分层李群，或p-演化方程高阶运营商的R n或组，结果已经是新的在所有这些情况下。
In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of “very weak solutions” to the Cauchy problem, that was already successfully used in similar contexts in [12] and [20]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p-evolution equations for higher order operators on R n or on groups, the results already being new in all these cases.
影响因子：
2.5
作者：
Garetto C
通讯作者：
Garetto C