关于色散Burgers型方程的柯西问题
偏微分方程分析
2025-10-13 v2
摘要
我们研究拟线性化的弱色散Burgers型方程：∂ t u + T u ∂ x u + ∂ x ∣ D ∣ α − 1 u = 0 , α ∈ ] 1 , 2 [ , \partial_t u+T_u \partial_xu+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[, ∂ t u + T u ∂ x u + ∂ x ∣ D ∣ α − 1 u = 0 , α ∈ ] 1 , 2 [ , ∂ t u + u ∂ x u + ∂ x ∣ D ∣ α − 1 u = 0 , α ∈ ] 1 , 2 [ , \partial_t u+u\partial_x u+\partial_x |D|^{\alpha-1}u=0,\ \alpha \in ]1,2[, ∂ t u + u ∂ x u + ∂ x ∣ D ∣ α − 1 u = 0 , α ∈ ] 1 , 2 [ , u 0 ∈ H s ( D ) u_0 \in H^s({\mathbb D}) u 0 ∈ H s ( D ) D = T 或 R {\mathbb D}={\mathbb T} \text{ 或 } {\mathbb R} D = T 或 R H s ( D ) H^s({\mathbb D}) H s ( D ) ∥ D 2 − α ( u 2 ) ∥ L t 1 L x ∞ \left\Vert D^{2-\alpha}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x} D 2 − α ( u 2 ) L t 1 L x ∞ ∥ ∂ x u ∥ L t 1 L x ∞ \left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x} ∥ ∂ x u ∥ L t 1 L x ∞ α ∈ ] 2 , 3 [ \alpha\in ]2,3[ α ∈ ] 2 , 3 [ ∂ t [ T e i T p ( u ) u ] + ∂ x ∣ D ∣ α − 1 [ T e i T p ( u ) u ] = T R ( u ) u , α ∈ ] 2 , 3 [ , \partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{\alpha-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ \alpha \in ]2,3[, ∂ t [ T e i T p ( u ) u ] + ∂ x ∣ D ∣ α − 1 [ T e i T p ( u ) u ] = T R ( u ) u , α ∈ ] 2 , 3 [ , T p ( u ) T_{p(u)} T p ( u ) T R ( u ) T_{R(u)} T R ( u ) u ∈ L t ∞ C ∗ ( 2 − α ) + u\in L^\infty_t C^{(2-\alpha)^+}_* u ∈ L t ∞ C ∗ ( 2 − α ) +
引用
@article{arxiv.2103.03588,
title = {On the Cauchy problem of dispersive Burgers type equations},
author = {Ayman Rimah Said},
journal= {arXiv preprint arXiv:2103.03588},
year = {2025}
}
备注
Updated version after review which closely follows the journal version to appear in Indiana University Mathematics Journal, 2022. arXiv admin note: text overlap with arXiv:2103.03576