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A current based approach for the uniqueness of the continuity equation (2402.10719v1)
Published 16 Feb 2024 in math.AP
Abstract: We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u \in [L1 (0,T; W{1,p}(\mathbb{T}d)) \cap L\infty ((0,T) \times \mathbb{T}d)]d$ with $\operatorname{div}(u) - \in L1 (0,T; L\infty (\mathbb{T}d))$ and an initial datum $\rho_0 \in Lq (\mathbb{T}d)$, where $\mathbb{T}d$ is the $d$-dimensional torus and $ 1 \leq p,q \leq +\infty$ such that $1/p + 1/q =1$ without using the theory of renormalized solutions. We propose a more geometric approach which will however still rely on a strong $L1$ estimate on the commutator (which is the key technical tool when using renormalized solutions, too), but other than that will be based on the theory of currents.