论文标题
一类负阶分散方程中的周期性霍尔德波
Periodic Hölder waves in a class of negative-order dispersive equations
论文作者
论文摘要
我们证明了在形式的一类负面负阶分散方程中，具有精确$α$-Hölder的连续性的最高，垂us，周期性的周期性波动解决方案的存在\ begin \ begin {equation*} u_t +（| \ mathrm {| \ mathrm {d} |^{d} |^{ - α} { - α} u + n（U + n（u + n（U +）） α\ in（0，1）$带有均匀的傅立叶乘数$ | \ mathrm {d} |^{ - α} $。我们处理类型$ |的非线性$ n（u）$ | u |^p $或$ u | u |^{p -1} $对于所有真实的$ p> 1 $，并证明当$ n $奇怪时，波浪还具有反对称性，因此包含倒置尖端。工具涉及与分析全局分叉同时进行详细估计的详细估计，在此我们通过正规化解决了非平滑$ n $的问题。我们认为，在这种情况下，最高的反对称波的构建以及非平滑术语的正规化都是新的，在这种情况下，直接适用于Whitham的广义版本，Burgers，Burgers-Poisson，Poisson，Burgers，burgers，burgers--Hilbert--Hilbert，Hilbert，hilbert，degasperis-procesi，degasperis-procesi，降低的OStrovsky和BiDiclection and bidiriclection。
We prove the existence of highest, cusped, periodic travelling-wave solutions with exact and optimal $ α$-Hölder continuity in a class of fractional negative-order dispersive equations of the form \begin{equation*} u_t + (| \mathrm{D} |^{- α} u + n(u) )_x = 0 \end{equation*} for every $ α\in (0, 1) $ with homogeneous Fourier multiplier $ | \mathrm{D} |^{ - α} $. We tackle nonlinearities $ n(u) $ of the type $ | u |^p $ or $ u | u |^{p - 1} $ for all real $ p > 1 $, and show that when $ n $ is odd, the waves also feature antisymmetry and thus contain inverted cusps. Tools involve detailed pointwise estimates in tandem with analytic global bifurcation, where we resolve the issue with nonsmooth $ n $ by means of regularisation. We believe that both the construction of highest antisymmetric waves and the regularisation of nonsmooth terms to an analytic bifurcation setting are new in this context, with direct applicability also to generalised versions of the Whitham, the Burgers--Poisson, the Burgers--Hilbert, the Degasperis--Procesi, the reduced Ostrovsky, and the bidirectional Whitham equations.
