Propagation of Sobolev Regularity for Integral-Invariant Weak Solutions of the Linear Transport Equation

Previous work strengthened the time integrability of integral-invariant weak solutions from L^1 to L^\infty within a spatial L^2 framework.This paper studies the propagation of Sobolev regularity for integral-invariant weak solutions of the conservative linear scalar transport equation.We first derive a representation formula along the flow and a distributional equation from integral identities against adjoint-transported test functions on domains moving with the same characteristic flow,and relate the weak formulation in time, including an endpoint term for the initial data, to representatives satisfying the pointwise-in-time integral definition.We then give a unified proof for k=1,2:under u_0\in H^k(Ω_0)and Assumption \mathsf A_{k+1},the pullback of the solution belongs to C^1([0,T];H^k(Ω_0)),the material derivative has spatial $H^k$ regularity,and the Eulerian time derivative has spatial H^{k-1} regularity.The corresponding time continuity is understood after pullback to the fixed domain by the flow.In both cases, U\in H^1(Q);when k=2, we further obtain \nabla_xU\in H^1(Q;\R^d).The pullback differentiability and the spatial estimates for the material derivativeprovide information beyond first-order space-time Sobolev regularity alone.The proof combines estimates for the characteristic flow with Sobolev product and composition rules.Under the additional assumption that the zero extension of the initial data belongs to H^2(\R^d),we also establish $H^1(0,T;H^1(\R^d))$ regularity in the fixed ambient space.Counterexamples show that these results concern regularity propagation under additional assumptions: general L^2 initial data do not automatically yield spatial smoothing,and spatial regularity of the material derivative cannot replace the corresponding regularity of the Eulerian time derivative.