[Paper Review] $L^2$ well posed Cauchy Problems and Symmetrizability of First Order Systems

Guy Métivier|arXiv (Cornell University)|Oct 3, 2013

Advanced Mathematical Physics Problems17 references3 citations

TL;DR

This paper establishes that Lipschitz continuity in $(t,x, au)$ of a microlocal symmetrizer is sufficient for $L^2$ well-posedness of first-order systems, and proves that the existence of symmetrizers with a given smoothness is equivalent to the existence of full symmetrizers with the same smoothness. This equivalence enables the invariance of symmetrizer existence under time reversal, ensuring local uniqueness and finite speed of propagation.

ABSTRACT

The Cauchy problem for first order system $L(t, x, \\D_t, \\D_x)$ is known to be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \\xi)$ which is smooth in $\\xi$ and Lipschitz continuous in $(t, x)$. This paper contains three main results. First we show that a Lipsshitz smoothness globally in $(t,x, \\xi)$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of \\emph{full symmetrizers} having the same smoothness. This notion was first introduced in \\cite{FriLa1}. This is the key point to prove the third result that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.

Motivation & Objective

• To establish that Lipschitz continuity in $(t,x, au)$ of a microlocal symmetrizer is sufficient for $L^2$ well-posedness of first-order systems.
• To prove the equivalence between the existence of symmetrizers with a given smoothness and the existence of full symmetrizers with the same smoothness.
• To show that the existence of microlocal symmetrizers is invariant under time reversal, implying local uniqueness and finite speed of propagation.
• To clarify the role of strong hyperbolicity and symmetrizer smoothness in the well-posedness of hyperbolic systems beyond the symmetric case.

Proposed method

• Uses wave packet localization and microlocal analysis to derive energy estimates via a main estimate involving the symmetrizer $S(t,x, au)$.
• Applies para-differential calculus and functional calculus to handle systems with limited regularity in coefficients.
• Establishes equivalence between microlocal symmetrizers and full symmetrizers by analyzing spectral projectors and eigenvalue continuity.
• Employs a homotopy argument over $t \in [0,1]$ to preserve positivity of the symmetrizer along paths of matrices $J_t$.
• Relies on the boundedness and symmetry of $\mathbf{S}(\tau,a)$ and the uniform positivity of $\mathrm{Re}(\mathbf{S}J u,u)$ on the kernel of $L(\tau,a)$.
• Constructs a global symmetrizer $S(a)$ via spectral decomposition: $S(a) = \sum_{\tau \in \Sigma(a)} \Pi(\tau,a)^* \mathbf{S}(\tau,a) \Pi(\tau,a)$, ensuring symmetry and positivity.

Experimental results

Research questions

• RQ1 Is Lipschitz continuity in $(t,x,\tau)$ of a microlocal symmetrizer sufficient for $L^2$ well-posedness of first-order systems?
• RQ2 Does the existence of a symmetrizer with a given smoothness imply the existence of a full symmetrizer with the same smoothness?
• RQ3 Is the existence of microlocal symmetrizers preserved under time reversal, and what are the implications for local uniqueness and finite speed of propagation?
• RQ4 Can the symmetrizer construction be extended to systems with non-constant multiplicities or generic double eigenvalues?
• RQ5 What is the precise relationship between strong hyperbolicity of the symbol and the existence of symmetrizers with limited regularity?

Key findings

• Lipschitz continuity in $(t,x,\tau)$ of the microlocal symmetrizer $S(t,x,\tau)$ is sufficient for $L^2$ well-posedness of the Cauchy problem.
• The existence of symmetrizers with a given smoothness (e.g., continuous, Lipschitz, $C^\infty$) is equivalent to the existence of full symmetrizers with the same smoothness.
• The existence of microlocal symmetrizers is invariant under time reversal, which implies local uniqueness and finite speed of propagation of singularities.
• For systems with generic double eigenvalues, smooth symmetrizers exist under strong hyperbolicity, and the Lipschitz condition is sharp—counterexamples exist for non-Lipschitz symmetrizers.
• The symmetrizer $S(a)$ constructed via spectral decomposition is continuous (resp. Lipschitz, $C^\infty$) in $a$ if $\mathbf{S}(\tau,a)$ and the spectral projectors are continuous (resp. Lipschitz, $C^\infty$).
• The positivity condition $\mathrm{Re}(\mathbf{S}J u,u) \geq c|u|^2$ on $\ker L(\tau,a)$ ensures uniform boundedness and uniform positivity of the constructed symmetrizer $S(a)$.