论文标题
广义本杰明方程的分析半径的演变
Evolution of the radius of analyticity for the generalized Benjamin equation
论文作者
论文摘要
在这项工作中，我们考虑了广义本杰明方程的初始价值问题 \ begin {qore} \ label {benj-ivp} \ begin {case} \ partial_t u-l \ mathcal {h} \ partial_x^2U- \ partial_x^3U+u^p p \ u^p \ partial_xu = 0， t \ in \ mathbb {r}; \; \;，\; p \ geq 1，\\ u（x，0）= u_0（x），\ end {cases} \ end {equation}其中$ u = u（x，x，t）$是一个真正的值函数，$ 0 <l <1 $ and $ \ mathcal {h} $是Hilbert Transform。该模型是由T. B. Benjamin（J. FluidMech。245（1992）401--411）引入的，并描述了在两流体系统中长波的单向传播，其中具有较高密度的较低流体无限深度深，界面是毛细管的。 我们证明，在围绕真实轴周围的条带上的函数分析空间中给定数据相关的IVP的局部解决方案继续进行分析，而不会及时缩小条带的宽度。我们还研究了空间分析性半径时间的演变，并表明它可以随着时间的推移而减小。最后，我们在空间分析性均匀半径的时间内可能会降低代数下限。
In this work we consider the initial value problem for the generalized Benjamin equation \begin{equation}\label{Benj-IVP} \begin{cases} \partial_t u-l\mathcal{H} \partial_x^2u-\partial_x^3u+u^p\partial_xu = 0, \quad x,\; t\in \mathbb{R};\;\;,\; p\geq 1, \\ u(x,0) = u_0(x), \end{cases} \end{equation} where $u=u(x,t)$ is a real valued function, $0<l<1$ and $\mathcal{H}$ is the Hilbert transform. This model was introduced by T. B. Benjamin (J. Fluid Mech. 245 (1992) 401--411) and describes unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We prove that the local solution to the IVP associated with the generalized Benjamin equation for given data in the spaces of functions analytic on a strip around the real axis continue to be analytic without shrinking the width of the strip in time. We also study the evolution in time of the radius of spatial analyticity and show that it can decrease as the time advances. Finally, we present an algebraic lower bound on the possible rate of decrease in time of the uniform radius of spatial analyticity.
