Pith Number

pith:CNNFGHD3

pith:2026:CNNFGHD3I5Z4IHHIB5WLHZZK2V

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Existence and regularity of solutions to parabolic-elliptic nonlinear systems

Marco Picerni

Existence of solutions to the parabolic-elliptic system is established for L1 data, with u gaining summability in L^s and L^q W^{1,q} despite the coupling term being only L2.

arxiv:2604.16100 v2 · 2026-04-17 · math.AP

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Claims

C1strongest claim

We prove existence results for data f∈L¹(Ω_T) and a corresponding increase in summability that obeys the L^p-regularity theorems for parabolic equations proved by Aronson-Serrin and by Boccardo-Dall'Aglio-Gallouët-Orsina. In particular, despite the term u M(x)∇ψ not being regular enough (since it only belongs to L²(Ω_T)), the solution u belongs to L^s(Ω_T)∩L^q(0,T;W^{1,q}_0(Ω)) for suitable s>1 and q>1.

C2weakest assumption

The coefficients A(x,t) and M(x) satisfy the standard measurability, boundedness and uniform ellipticity conditions required to invoke the cited parabolic regularity theorems; these assumptions are implicit in the abstract but not stated explicitly.

C3one line summary

Existence and higher summability are shown for solutions of a parabolic-elliptic system with discontinuous coefficients, L^1 data, and |u|^θ nonlinearity where θ < 2/N.

Receipt and verification

First computed	2026-05-22T01:03:19.324993Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

135a531c7b4773c41ce80f6cb3e72ad551f50e5f2cedceda6ac09e54f995d14f

Aliases

arxiv: 2604.16100 · arxiv_version: 2604.16100v2 · doi: 10.48550/arxiv.2604.16100 · pith_short_12: CNNFGHD3I5Z4 · pith_short_16: CNNFGHD3I5Z4IHHI · pith_short_8: CNNFGHD3

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CNNFGHD3I5Z4IHHIB5WLHZZK2V \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 135a531c7b4773c41ce80f6cb3e72ad551f50e5f2cedceda6ac09e54f995d14f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "ccbed6e5c76e9d8e064b02d7a3c8fa1b755a6c34fa1bc02eb4281371ac446d69",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-04-17T14:34:04Z",
    "title_canon_sha256": "b8233eb8d4a83fdd7baf5b99471b6bd052923abc5a845a728313d1bb4a83c224"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.16100",
    "kind": "arxiv",
    "version": 2
  }
}