[Paper Review] Inhomogeneous Strichartz estimates with spherical symmetry and applications to the Dirac-Klein-Gordon system in two space dimensions

Evgeni Y. Ovcharov|ArXiv.org|Mar 30, 2009

Advanced Mathematical Physics Problems18 references3 citations

TL;DR

This paper establishes inhomogeneous Strichartz estimates with spherical symmetry for the wave equation in two space dimensions using duality arguments, enabling global existence of spherically symmetric solutions to the Dirac-Klein-Gordon system for initial data in low regularity spaces, specifically with $1/4 < r < 1/2$. The key contribution is a novel definition of spherical symmetry for spinors via rotation-equivariance, which preserves the system's structure and allows the use of radial estimates in the energy-critical regime.

ABSTRACT

In this {\bf draft version} we prove inhomogeneous Strichartz estimates with spherical symmetry in the abstract setting via duality arguments. Then we derive some new explicit estimates in the context of the wave equation. This allows us to prove global existence of spherically symmetric solutions to the Dirac-Klein-Gordon (DKG) system in two space dimensions.

Motivation & Objective

• To develop inhomogeneous Strichartz estimates with spherical symmetry in the abstract setting using duality, extending known homogeneous estimates.
• To define a consistent notion of spherical symmetry for spinors in two space dimensions, accounting for the action of rotations on spinor space.
• To apply these estimates to prove global existence of spherically symmetric solutions to the Dirac-Klein-Gordon system in two space dimensions.
• To address the global well-posedness of the DKG system in low regularity regimes where standard energy methods fail due to lack of positive definite energy.

Proposed method

• Use duality arguments to derive inhomogeneous Strichartz estimates from homogeneous ones, leveraging the isometry between $L^p(\mathbb{R}^n)$ functions with spherical symmetry and $L^p([0,\infty); \rho^{n-1}d\rho)$ functions.
• Define spherical symmetry for spinors as invariance under the unitary action of rotations on spinor space, ensuring consistency with the Dirac operator's transformation properties.
• Apply the derived estimates to the inhomogeneous wave equation with radial data, obtaining control in mixed-norm spaces with sharp regularity gain.
• Use the charge conservation of the Dirac field and the radial Strichartz estimates to close a contraction argument in the function space framework.
• Establish a priori bounds in $L^2_t L^2_x$ for the spinor and $H^r_t H^r_x$ for the scalar field, relying on the null-form structure of the nonlinearities.
• Prove existence and uniqueness of global solutions in the class $\psi \in C((0,\infty); L^2), \phi \in C((0,\infty); H^r) \cap C^1((0,\infty); H^{r-1})$ for $1/4 < r < 1/2$.

Experimental results

Research questions

• RQ1 Can inhomogeneous Strichartz estimates with spherical symmetry be derived from homogeneous ones using duality in the radial setting?
• RQ2 How should spherical symmetry be defined for spinors in two space dimensions to preserve the structure of the Dirac equation under rotations?
• RQ3 What is the sharp regularity threshold for global existence of spherically symmetric solutions to the Dirac-Klein-Gordon system in two space dimensions?
• RQ4 Can the null-form structure of the DKG system's nonlinearities be exploited in conjunction with radial Strichartz estimates to achieve low-regularity well-posedness?
• RQ5 Does the charge conservation of the Dirac field allow for global existence results in the absence of a positive definite energy?

Key findings

• The paper establishes inhomogeneous Strichartz estimates with spherical symmetry for the wave equation in two space dimensions, with a gain of regularity in the radial setting.
• A new definition of spherical symmetry for spinors is introduced via rotation-equivariance, ensuring compatibility with the Dirac operator's transformation under spatial rotations.
• The system's null-form structure allows for global existence of solutions even when only $L^2$-boundedness is available for the spinor, without a positive definite energy.
• Global existence of spherically symmetric solutions to the Dirac-Klein-Gordon system is proven for initial data in $L^2 \times H^r \times H^{r-1}$ with $1/4 < r < 1/2$, extending previous local results.
• The solution class is shown to be unique and to depend continuously on the initial data in the specified norms.
• The proof relies on a duality-based derivation of radial inhomogeneous estimates and the use of charge conservation to control the spinor norm over time.