[Paper Review] A Strichartz estimate for de Sitter space

Dean Baskin|arXiv (Cornell University)|Sep 14, 2009

Advanced Mathematical Physics Problems11 references3 citations

TL;DR

This paper establishes global Strichartz estimates for the conformally invariant Klein-Gordon equation on $C^2$ asymptotically de Sitter spaces satisfying a short-range condition, using a rescaling argument and local Strichartz estimates. The key result is a family of $L^pL^q$ estimates with exponential weights that enable well-posedness for semilinear Klein-Gordon equations with small data on these spacetimes.

ABSTRACT

We demonstrate a family of Strichartz estimates for the conformally invariant Klein-Gordon equation on a class of asymptotically de Sitter spaces with C^2 metrics by using well-known local Strichartz estimates and a rescaling argument. This class of metrics includes de Sitter space. We also give an application of the estimates to a semilinear Klein-Gordon equation on these spaces.

Motivation & Objective

• To derive global Strichartz estimates for the conformally invariant Klein-Gordon equation on asymptotically de Sitter spaces with $C^2$ metrics.
• To extend local Strichartz estimates to global estimates using a rescaling argument and conformal compactification.
• To prove well-posedness for semilinear Klein-Gordon equations with small initial data on these spacetimes.
• To demonstrate that the absence of dispersive obstructions for $oxed{\lambda = \frac{n^2-1}{4}}$ allows global Strichartz estimates.

Proposed method

• Use conformal compactification of de Sitter space to a Lorentzian cylinder, transforming the problem into a setting where local Strichartz estimates apply.
• Apply a rescaling argument to convert local estimates into global estimates with exponential weights in time.
• Conjugate the Klein-Gordon operator to a form amenable to time-dependent Strichartz estimates from Tataru and Smith.
• Use the short-range assumption (no linear term in Taylor expansion of $h(x,y,dy)$ at $x=0$) to control error terms in the rescaling.
• Translate estimates from compactified $(x,y)$ coordinates to asymptotic $(t,y)$ coordinates via $x = e^{-t}$.
• Apply a contraction mapping argument in weighted $L^pL^q$ spaces to prove existence and uniqueness of small data solutions to the semilinear equation.

Experimental results

Research questions

• RQ1 Can global Strichartz estimates be established for the conformally invariant Klein-Gordon equation on asymptotically de Sitter spaces with $C^2$ metrics?
• RQ2 Under what conditions does the absence of dispersive obstructions allow global Strichartz estimates?
• RQ3 Can the resulting estimates be used to prove small data well-posedness for semilinear Klein-Gordon equations on these spacetimes?
• RQ4 How does the conformal method facilitate the extension of local Strichartz estimates to global ones in this geometric setting?

Key findings

• Global Strichartz estimates hold for the conformally invariant Klein-Gordon equation on $C^2$ asymptotically de Sitter spaces satisfying the short-range condition (no linear term in $h(x,y,dy)$).
• The estimates are of the form $\|u\|_{L^p_t(W^{1-s,q}_y(e^{n|t|}dh), e^{p(s-1/2)|t|}dt)} \lesssim e^{|t_0|/2}(\|u_0\|_{H^1(e^{n|t|}dh)} + \|u_1\|_{L^2(e^{n|t|}dh)})$ for admissible $(p,q,s)$.
• For the semilinear equation $\Box_g u + \frac{n^2-1}{4}u = F_k(u)$, a unique small data solution exists in $L^5_t(L^{10}_y(e^{3|t|}dh), e^{5|t|/2}dt)$ when $n=3$, $k=5$, and initial data are sufficiently small.
• A similar small data solution exists in $L^3_t(L^6_y(e^{4|t|}dh), e^{3|t|/2}dt)$ for $n=4$, $k=3$, with the same smallness condition on initial data.
• The solution depends Lipschitz continuously on the initial data, and uniqueness follows from a contraction argument using the $L^1_t(L^2_y)$ norm of the nonlinearity.
• The method extends to the full de Sitter space and to a class of asymptotically de Sitter spaces, improving on prior results that were limited to static models.