2020/12/28 by Hartmut Pecher, Pecher, Hartmut
Engineering · Mathematics · Physics and Astronomy · #35L70 #35Q40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2012.14239
openalex publication_date 2020/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The local well-posedness problem for the Maxwell-Klein-Gordon system in\nCoulomb gauge as well as Lorenz gauge is treated in two space dimensions for\ndata with minimal regularity assumptions. In the classical case of data in\nL2-based Sobolev spaces Hs and Hl for the electromagnetic field \φ\nand the potential A, respectively. The minimal regularity assumptions are s\n> \(1)/(2) and l > \(1)/(4) , which leaves a gap of \(1)/(2) and\n\(1)/(4) to the critical regularity with respect to scaling sc = lc =0\n. This gap can be reduced for data in Fourier-Lebesgue spaces\n widehatHs,r and widehatHl,r to s> \(21)/(16) and l >\n\(9)/(8) for r close to 1 , whereas the critical exponents with respect\nto scaling fulfill sc \→ 1 , lc \→ 1 as r \→ 1 . Here\n\‖f\‖_ widehatHs,r := \‖ \⟨ \ξ \⟩s\n\f\‖_Lr'\τ \ξ , , , 1 < r \≤ 2 , , ,\n\(1)/(r)+\(1)/(r') = 1 , . Thus the gap is reduced for \φ as well\nas A in both gauges.\n