[Paper Review] Gevrey well posedness for $3$-evolution equations with variable coefficients

Alexandre Arias, Alessia Ascanelli+1 more|arXiv (Cornell University)|Jun 17, 2021

Advanced Mathematical Physics Problems26 references4 citations

TL;DR

This paper establishes Gevrey well-posedness for the Cauchy problem associated with third-order anisotropic evolution equations with variable coefficients, including complex-valued lower-order terms. By employing pseudodifferential operator theory and Fourier multiplier estimates in Gevrey-type spaces, the authors prove existence, uniqueness, and stability of solutions in the space $\mathcal{H}^\infty_\theta(\mathbb{R})$, extending the theory of $p$-evolution equations to the $p=3$ case in the Gevrey setting for the first time.

ABSTRACT

We study the Cauchy problem for a class of third order linear anisotropic evolution equations with complex valued lower order terms depending both on time and space variables. Under suitable decay assumptions for $|x| o \infty$ on these coefficients, we prove a well posedness result in Gevrey-type spaces.

Motivation & Objective

• To extend the theory of $p$-evolution equations to the case $p=3$ in Gevrey-type spaces, where only $p=2$ (Schrödinger-type) results were previously known.
• To analyze the well-posedness of third-order linear anisotropic evolution equations with complex-valued, variable-coefficient lower-order terms.
• To establish sufficient decay conditions on the lower-order coefficients as $|x|\to\infty$ to ensure well-posedness in Gevrey classes.
• To provide a foundational linear theory for future study of semilinear $3$-evolution equations using Nash-Moser methods.

Proposed method

• The authors use a functional framework based on Gevrey-type Sobolev spaces $H^m_{\rho;\theta}(\mathbb{R})$, defined via Fourier multipliers involving $\langle D\rangle^m e^{\rho\langle D\rangle^{1/\theta}}$, to capture Gevrey regularity.
• They introduce symbol classes $S^m_{\mu,\nu}(\mathbb{R}^{2n};A)$ to control the growth and regularity of pseudodifferential operators with Gevrey symbols.
• A key step involves constructing a parametrix for the operator $P(t,x,D_t,D_x)$ using oscillatory integral representations and microlocal analysis.
• The proof relies on estimating remainder terms via integration by parts and exponential decay estimates, particularly for the regularizing terms $r_{1,N'}, r_{2,N'}, r_{3,N'}$, using phase function analysis and complex deformation techniques.
• The authors analyze the behavior of the operator $e^{-\Lambda}(x,D)$ and its adjoint to control the growth of solutions in Gevrey norms.
• They establish that the composition $r_\infty(x,D) \circ \{e^{-\Lambda}(x,D)\}^*$ is a regularizing operator, ensuring the parametrix construction is well-behaved.

Experimental results

Research questions

• RQ1 Can the well-posedness theory for $p$-evolution equations be extended from $p=2$ to $p=3$ in Gevrey-type spaces?
• RQ2 What decay conditions on variable coefficients (especially complex-valued lower-order terms) are necessary and sufficient for Gevrey well-posedness in the $3$-evolution case?
• RQ3 How can the parametrix method be adapted to handle third-order anisotropic operators with variable coefficients in the Gevrey setting?
• RQ4 Can the linear theory for $3$-evolution equations support future analysis of semilinear problems via Nash-Moser methods?
• RQ5 What role does the complex phase deformation play in achieving exponential decay estimates for error terms in the parametrix construction?

Key findings

• The Cauchy problem for the $3$-evolution operator $P(t,x,D_t,D_x)$ is well-posed in the space $\mathcal{H}^\infty_\theta(\mathbb{R})$, which lies between compactly supported Gevrey functions and the full Gevrey class $G^\theta(\mathbb{R})$.
• Well-posedness is established under decay assumptions on the lower-order coefficients as $|x|\to\infty$, which are essential for controlling growth in the Gevrey framework.
• The remainder terms $r_{1,N'}, r_{2,N'}, r_{3,N'}$ in the parametrix construction are shown to satisfy exponential decay estimates, ensuring the parametrix is well-defined and bounded in Gevrey norms.
• The composition $r_\infty(x,D) \circ \{e^{-\Lambda}(x,D)\}^*$ is proven to be a regularizing operator, confirming the smoothing nature of the error terms.
• The method successfully generalizes the $p=2$ case to $p=3$, providing the first well-posedness result in Gevrey spaces for third-order evolution equations with variable coefficients.
• The results lay a rigorous foundation for future study of semilinear $3$-evolution equations using Nash-Moser techniques in the Gevrey setting.