arXiv 2103.035882021-03math.AP

色散Burgers型方程的Cauchy问题

On the Cauchy problem of dispersive Burgers type equations

Ayman Rimah Said

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
发表信息：: Indiana University Mathematics Journal, 2023
DOI：: 10.1512/iumj.2023.72.9409

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AI总结：

本文研究弱色散Burgers型方程的Cauchy问题，通过抛物复Cole-Hopf变换改进先验估计，消除波破缺，并在α∈(2,3)时将线性化方程共轭为半线性方程。

摘要：

我们研究线性化弱色散Burgers型方程：$$\partial_t u+T_u \partial_xu+\partial_x |D|^{α-1}u=0,\\ α\in ]1,2[,$$ 该方程包含通常弱色散Burgers型方程：\\[ \partial_t u+u\partial_x u+\partial_x |D|^{α-1}u=0,\\ α\in ]1,2[, \\] 的主要非线性“最坏相互作用”项，即低-高相互作用项，其中$u_0 \in H^s({\mathbb D})$，${\mathbb D}={\mathbb T} \text{ 或 } {\mathbb R}$。通过我们在[42]中引入的抛物复Cole-Hopf型规范变换，我们在$\left\Vert D^{2-α}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x}$的控制下证明了$H^s({\mathbb D})$中一个新的先验估计，改进了通常的双曲控制$\left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x}$。因此，我们消除了[31]中推测的爆破解的“标准”波破缺情形。对于$α\in ]2,3[$，我们证明可以将线性化色散Burgers方程完全共轭为如下半线性方程：$$\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{α-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\\ α\in ]2,3[,$$ 其中$T_{p(u)}$和$T_{R(u)}$是定义在$u\in L^\infty_t C^{(2-α)^+}_*$上的0阶抛物微分算子。

英文摘要：

We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+T_u \partial_xu+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[,$$ which contains the main non linear "worst interaction" terms, that is low-high interaction terms, of the usual weakly dispersive Burgers type equation: \[ \partial_t u+u\partial_x u+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[, \] with $u_0 \in H^s({\mathbb D})$, where ${\mathbb D}={\mathbb T} \text{ or } {\mathbb R}$. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [42], we prove a new a priori estimate in $H^s({\mathbb D})$ under the control of $\left\Vert D^{2-α}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x}$, improving upon the usual hyperbolic control $\left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x}$. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured in [31]. For $α\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{α-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ α\in ]2,3[,$$ where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L^\infty_t C^{(2-α)^+}_*$.

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