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17 source collections · 43 mathematical fields

Open problemEditorial · Partial differential equations

Navier-Stokes existence and smoothness

For three-dimensional incompressible flow, determine whether $\partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\Delta u$ with $\nabla\cdot u=0$ always has a global smooth solution, or exhibit finite-time breakdown.

Source checked Jul 24, 20261 pinned Lean statementInspect problem

Source labels openMillennium Problems · Partial differential equations

Existence And Smoothness Of The Navier–Stokes Equation

(A) Existence and smoothness of Navier–Stokes solutions on ℝ³.

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openMillennium Problems · Partial differential equations

Existence And Smoothness Of The Navier–Stokes Equation

(B) Existence and smoothness of Navier–Stokes solutions in ℝ³/ℤ³.

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openMillennium Problems · Partial differential equations

Existence And Smoothness Of The Navier–Stokes Equation

(C) Breakdown of Navier–Stokes solutions on ℝ³.

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openMillennium Problems · Partial differential equations

Existence And Smoothness Of The Navier–Stokes Equation

(D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³.

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openPapers · Partial differential equations

De Giorgi's conjecture

De Giorgi's conjecture holds in dimension $n ≤ 8$ .

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openPapers · Partial differential equations

De Giorgi's conjecture

De Giorgi's conjecture holds in dimension $n = 4$ .

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openPapers · Partial differential equations

De Giorgi's conjecture

De Giorgi's conjecture holds in dimension $n = 5$ .

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openPapers · Partial differential equations

De Giorgi's conjecture

De Giorgi's conjecture holds in dimension $n = 6$ .

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openPapers · Partial differential equations

De Giorgi's conjecture

De Giorgi's conjecture holds in dimension $n = 7$ .

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openPapers · Partial differential equations

De Giorgi's conjecture

De Giorgi's conjecture holds in dimension $n = 8$ .

Source checked Jul 26, 20261 pinned Lean statementInspect problem