Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations

We show that solutions to the modified Dirac-Klein-Gordon system in standard notation {Mathematical expression} in two space dimensions with complex-valued initial data ψ(0,x)∈L² (ℝ²;ℂ⁴), real valued φ{symbol}(0, x) ∈ W^1,2 (ℝ²) and φ{symbol}_t (0, x) ∈ L² (ℝ²) have regularity[Figure not available: see fulltext.] Here g{script}_loc¹ (ℝ³) denotes the (local) Hardy space, and g(t) is assumed to be in C¹(ℝ) and g(0)=0. Consequently nonlinear terms φψ which appear in the classical coupled Dirac-Klein-Gordon system (with the modification g=g(t)∈C¹ and g(0)=0) can then be defined in L_loc^∞ (ℝ²; L¹ (ℝ²)). We hope these results will be useful in establishing the existence of weak solutions to the classical coupled Dirac-Klein-Gordon system in the framework of compensated compactness.