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Lions in \\cite{Lions96} concerning the propagation of regularities of density patch for viscous inhomogeneous flow, we first establish the global in time well-posedness of two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density being of the form: $\\eta_1{\\bf 1}_{\\Om_0}+\\eta_2{\\bf 1}_{\\Om_0^c},$ for any pair of positive constants $(\\eta_1,\\eta_2),$ and for any bounded, simply connected $W^{k+2,p}(\\R^2)$ domain $\\Om_0.$ We then prove that the time evolved domain $\\Om(t)$ also belongs to the class of $W^{k+2,p}$ for any $t>"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.07922","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-27T03:48:37Z","cross_cats_sorted":[],"title_canon_sha256":"bfb0af7b1f3cc02b01c2c9c1cdbf2babb7f0586f1ad0003a933a1d19860edbdb","abstract_canon_sha256":"48df4926a65eb4d91c1f950ebe1482242e41d4606b757a9739db93fe3e5856af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:08.615239Z","signature_b64":"Ie0Ic1yK7y5UwzFO3Bfi2LQl/V+Y/zkfF3+EPMHYfoqNKaDFsQeHRBsZB5UTy05VRqT0BdPKtgPVbwtSC0c1Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73cdccfe7ee351ad3d25ccce30d66e655dc62cd32c934a7dbe85515d63d57866","last_reissued_at":"2026-05-18T01:16:08.614592Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:08.614592Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global regularities of two-dimensional density patch for inhomogeneous incompressible viscous flow with general density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ping Zhang, Xiao Liao","submitted_at":"2016-04-27T03:48:37Z","abstract_excerpt":"Toward the open question proposed by P.-L. 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