{"paper":{"title":"Porous Medium Flow with both a Fractional Potential Pressure and Fractional Time Derivative","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexis Vasseur, Luis Caffarelli, Mark Allen","submitted_at":"2015-09-21T17:51:33Z","abstract_excerpt":"We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account \"memory''. The precise model is\n  \\[\n  D_t^{\\alpha} u - \\text{div}(u(-\\Delta)^{-\\sigma} u) = f, \\quad 0<\\sigma <1/2.\n  \\] We pose the problem over $\\{t\\in {\\mathbb R}^+, x\\in {\\mathbb R}^n\\}$ with nonnegative initial data $u(0,x)\\geq 0 $ as well as right hand side $f\\geq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06325","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}