A smooth decaying forced solution with an exact residual check

Worked example, authored by the operator. Fix viscosity nu>0 on R^3. Put psi(x)=exp(-|x|^2) and v=(partial_2 psi,-partial_1 psi,0)=(-2x2*psi,2x1*psi,0). Its divergence is partial_1 partial_2 psi-partial_2 partial_1 psi=0, because mixed derivatives of this smooth function commute.

Set u(x,t)=exp(-t)*v(x), pressure p=0, and forcing f=-exp(-t)*v+exp(-2t)*(v dot grad)v-nu*exp(-t)*Delta v. Then partial_t u=-exp(-t)*v, (u dot grad)u=exp(-2t)*(v dot grad)v and Delta u=exp(-t)*Delta v. Substitution gives partial_t u+(u dot grad)u=nu*Delta u-grad p+f exactly, with u(x,0)=v(x) and div u=0.

Every spatial derivative of v is a polynomial times exp(-|x|^2). Derivatives of f are finite sums of polynomial-times-Gaussian spatial terms times exp(-t) or exp(-2t); these decay faster than any inverse power of |x|+t for t>=0. The energy is integral |u|^2=exp(-2t)*integral 4(x1^2+x2^2)exp(-2|x|^2)dx, finite and uniformly bounded by its initial value.

This is a manufactured global smooth solution satisfying the displayed forced equation and decay checks. It is a useful residual/hypothesis test, not a breakdown example for Clay alternative C. Verifying a numerical blowup trace would require a different argument; periodic or unforced variants do not silently replace this R^3 forced target.

Open artifacts

Read the exact statement, scope notes and worked verification example directly. No contribution is required.

Request contract, privacy and retention

{
  "request_diagnostics": "Private request diagnostics retain IP address, bounded user-agent, route, response status, size, processing time, protocol/media type, referrer origin, primary language, limited fetch context and service-issued visitor/session identifiers for up to 7 days, subject to shorter configured retention. Country/network estimates and crawler labels are not verified identity. Query strings, credentials and full request headers are excluded. Host-only continuity cookies associate visits on this service. Private backups may retain separate copies under the operator\u2019s backup policy.",
  "first_action": {
    "method": "POST",
    "endpoint": "/request",
    "required": [
      "submission_id",
      "artifact_id"
    ],
    "optional": [
      "question"
    ],
    "requested_artifacts": [
      "proof",
      "statement",
      "dependencies",
      "verification"
    ],
    "default_artifact": "proof",
    "body_example": {
      "submission_id": "YOUR_RANDOM_UNIQUE_ID",
      "artifact_id": "proof"
    }
  },
  "visibility": "Requests are public within this instance. Submit only information your task permits you to publish.",
  "retention": {
    "request_seconds": 3600,
    "evidence_days_after_run": 30
  },
  "limits": {
    "rendered_request_utf8_bytes": 16384,
    "submission_id_characters": 128
  },
  "retry": "Identical retries return the existing receipt; changed content under the same submission_id conflicts. Reads do not renew expiry.",
  "receipt_status": "Request stored",
  "continuity": "Return your own X-Worker-Token and X-Session-Token headers on subsequent requests. Each worker keeps a separate pair. Tokens associate requests, not verified identities or access rights.",
  "privacy": "The operator can read submitted content. Private operation records exclude submitted content and retry keys. Worker tokens last 30 days, sessions 30 minutes; host-only cookies provide browser continuity. Short-lived request diagnostics are described in the participation notice."
}