[Paper Review] A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calderón

Adrian Nachman, Idan Regev+1 more|arXiv (Cornell University)|Aug 16, 2017

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper establishes a nonlinear Plancherel theorem for the two-dimensional scattering transform arising in the inverse scattering method for the defocusing Davey-Stewartson II (DSII) equation. By proving $ L^2 $-boundedness of d-bar operators with non-smooth symbols and new estimates on fractional integrals, the authors achieve global well-posedness and scattering for DSII in $ L^2 $ for arbitrary initial data, and further apply the result to prove global uniqueness in the 2D Calderón inverse problem for conductivities with $ \log\sigma \in \dot{H}^1 $. The key contribution is a nonlinear analog of the Plancherel theorem that links scattering data to initial data in $ L^2 $.

ABSTRACT

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in $L^2$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension $2$, for conductivities $σ>0$ with $\log σ\in \dot H^1$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $L^2$-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

Motivation & Objective

• To establish a nonlinear Plancherel theorem for the scattering transform in two dimensions, specifically for the defocusing Davey-Stewartson II (DSII) equation.
• To prove global well-posedness and scattering in $ L^2 $ for the DSII equation with arbitrary initial data in $ L^2({\mathbb{R}}^2) $, overcoming the mass-critical nature of the problem.
• To extend the nonlinear Plancherel theorem to the inverse boundary value problem of Calderón in dimension 2, proving global uniqueness for conductivities $ \sigma > 0 $ with $ \log\sigma \in \dot{H}^1 $.
• To develop new estimates on classical fractional integrals and establish $ L^2 $-boundedness of pseudo-differential operators with non-smooth symbols in all dimensions.
• To provide a complete characterization of large-time behavior of solutions to DSII via the scattering transform, including completeness of wave operators.

Proposed method

• Prove a new $ L^2 $-boundedness result for $ \bar{\partial} $-operators with $ L^2 $-symbols, establishing uniform bounds depending only on the $ L^2 $-norm of the symbol.
• Use concentration-compactness and d-bar problem techniques to analyze the scattering transform and its invertibility in the $ L^2 $-critical setting.
• Establish a nonlinear Plancherel theorem by proving that the scattering transform is an isometry from $ L^2({\mathbb{R}}^2) $ to a weighted $ L^2 $-space of scattering data.
• Apply the nonlinear Plancherel theorem to the inverse scattering problem for DSII, proving global existence and scattering for all $ L^2 $ initial data.
• Reduce the Calderón problem to a d-bar problem with a non-smooth symbol via the transformation $ q = -\frac{1}{2}\partial\log\sigma $, and use the nonlinear Plancherel result to recover $ \log\sigma $ from the Dirichlet-to-Neumann map.
• Use layer potential methods and boundary integral equations to reduce the exterior d-bar problem to a boundary integral equation on $ \partial\Omega $, proving uniqueness of solutions under the $ \dot{H}^1 $-log condition.

Experimental results

Research questions

• RQ1 Can a nonlinear Plancherel theorem be established for the scattering transform associated with the defocusing Davey-Stewartson II equation in two dimensions?
• RQ2 Does the nonlinear Plancherel theorem imply global well-posedness and scattering in $ L^2 $ for the DSII equation with arbitrary initial data in $ L^2({\mathbb{R}}^2) $?
• RQ3 Can the nonlinear Plancherel theorem be extended to prove global uniqueness in the 2D inverse boundary value problem of Calderón for conductivities $ \sigma > 0 $ with $ \log\sigma \in \dot{H}^1 $?
• RQ4 What new estimates on fractional integrals and $ L^2 $-boundedness of pseudo-differential operators with non-smooth symbols are required to prove the nonlinear Plancherel theorem?
• RQ5 How can the scattering transform be used to characterize the large-time behavior of solutions to the DSII equation in the $ L^2 $-critical regime?

Key findings

• The authors prove that the scattering transform for the defocusing DSII equation is an isometry from $ L^2({\mathbb{R}}^2) $ to a weighted $ L^2 $-space of scattering data, establishing a nonlinear Plancherel theorem.
• Global well-posedness and scattering in $ L^2 $ for the DSII equation are established for all initial data in $ L^2({\mathbb{R}}^2) $, including large data, via the nonlinear Plancherel theorem.
• The solution to the DSII equation exhibits scattering behavior: as $ t \to \infty $, the solution approaches a free solution in $ L^2 $, with the wave operators being complete.
• The nonlinear Plancherel theorem implies global uniqueness in the 2D Calderón problem for conductivities $ \sigma > 0 $ with $ \log\sigma \in \dot{H}^1 $, meaning the Dirichlet-to-Neumann map uniquely determines $ \sigma $.
• The proof establishes a new $ L^2 $-boundedness result for pseudo-differential operators with non-smooth symbols, valid in all dimensions, which is crucial for handling the $ \bar{\partial} $-problem with $ L^2 $-symbols.
• The authors derive new estimates on classical fractional integrals, which are essential for controlling the nonlinear terms in the inverse scattering framework and for proving the stability of the scattering transform.