{"ID":23681139,"CreatedAt":"2026-09-18T08:49:17.088671394Z","UpdatedAt":"2026-09-18T08:49:17.088671394Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20682","arxiv_id":"2609.20682","title":"Discontinuity of the Vlasov--Poisson Flow in $L_x^pL_v^\\infty$","abstract":"We prove that solution maps of the Vlasov--Poisson equation are discontinuous at the zero initial datum in $X_p=L_x^pL_v^\\infty$ for every $1\\leq p\u003c\\infty$, in $\\mathbb{R}^{d} \\times \\mathbb{R}^{d}$ with $d \\le 3$, and for both attractive and repulsive interactions. For every $T\u003e0$, the trajectory of the solution in $L^\\infty([0,T];X_p)$ is discontinuous at zero, and for every sufficiently small fixed $t\u003e0$, the fixed-time map with values in $X_p$ is also discontinuous at zero. The counterexamples are smooth, nonnegative, bounded by one, and supported in a common compact subset of phase space. Their mass and initial $X_p$ norm tend to zero, whereas their solution norm at the observation time is at least one. The construction places narrow spatial packets on a lattice. Free transport allows a different velocity to select a packet at each spatial point, while a smallness estimate on the field ensures that the nonlinear characteristics are close to that of the free transport. For the same families, the phase-space $L^q$ norms converge to zero uniformly in time for every $1\\leq q\u003c\\infty$.","short_abstract":"We prove that solution maps of the Vlasov--Poisson equation are discontinuous at the zero initial datum in $X_p=L_x^pL_v^\\infty$ for every $1\\leq p\u003c\\infty$, in $\\mathbb{R}^{d} \\times \\mathbb{R}^{d}$ with $d \\le 3$, and for both attractive and repulsive interactions. For every $T\u003e0$, the trajectory of the solution in $L...","url_abs":"https://arxiv.org/abs/2609.20682","url_pdf":"https://arxiv.org/pdf/2609.20682v1","authors":"[\"Ke Chen\",\"In-Jee Jeong\",\"Quoc-Hung Nguyen\",\"Sangwook Tae\"]","published":"2026-09-17T16:51:40Z","proceeding":"math.AP","tasks":"[\"math.AP\"]","methods":"[]","has_code":false}