Well-posedness and Regularity of the Integral Invariant Model from Linear Scalar Transport Equation

An integral invariant model derived from the coupling of the transport equation and its adjoint equation is investigated.Despite extensive research on the numerical implementation of this model,no studies have yet explored the well-posedness and regularity of the model itself.To address this gap,firstly, a comprehensive mathematical definition is formulated as a Cauchy initial value problem for the integral invariant model.This formulation preserves essential background information derived from relevant numerical algorithms.In the above definition,we directly evolve the time-dependent test function $\psi(\mathbf{x},t)$ through explicit construction rather than solving the adjoint equation,which enables reducing the required regularity of the test function $\Psi(\mathbf{x})$ from $C^1(\Omega)$ to $L^2(\Omega)$,contributing to stability proof. The challenge arising from the mismatch of integration domains on both sides of the model's equivalent form is overcome through the compact support property of test functions. For any arbitrary time instant $t^{*}\in[0,T]$,an abstract function $\mathcal{U}(\lambda)$ taking values in the Banach space $L^2(\mathbb{R}^d)$ is initially constructed on the entire space $\mathbb{R}^d$ via the Riesz representation theorem.Subsequently,this function is properly restricted to the time-dependent bounded domain $\widetilde{\Omega}(t)$ through multiplication by the characteristic function. The existence of this model's solution in $L^{1}([0,T],L^2(\widetilde{\Omega}(t)))$ is then rigorously established.Furthermore,by judiciously selecting test functions $\Psi$, the stability of the integral invariant model is proved, from which the uniqueness naturally follows.Finally, when the initial value $\widetilde{U}_0 \in L^{2}(\widetilde{\Omega}(0))$,the temporal integrability of the model over $[0,T]$ can be enhanced to $L^{\infty}([0,T],L^2(\widetilde{\Omega}(t)))$.