L1 Stokes semigroup generation problem

Let $\Omega\subset\mathbb{R}^d$ be a bounded $\mathrm{C}^{1,\alpha}$ -domain, and let $A_{1,\sigma,n}$ denote the Stokes operator with no-slip boundary conditions on $\mathrm{L}^1_{\sigma,n}(\Omega)$ . Determine whether $A_{1,\sigma,n}$ generates a $\mathrm{C}_0$ -semigroup on $\mathrm{L}^1_{\sigma,n}(\Omega)$ . At the endpoint $p=1$ , also determine the corresponding generation behavior of the realization of the Stokes operator on $\mathrm{L}^1(\Omega,\mathbb{C}^d)/\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})$ , and whether this quotient realization is genuinely distinct from the usual solenoidal realization.

References

Primary source

arXiv

Additional references

The L1-Stokes Semigroup — arXiv

fluid-mechanics functional-analysis partial-differential-equations

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A preprint claims to settle the endpoint question by proving generation in a quotient setting while ruling it out for the usual setting, but this has not been independently verified.

The problem concerns whether the Stokes operator generates a continuous semigroup at the endpoint space $L^1$ . The retrieved preprint claims that the answer depends on using a quotient by gradients rather than the usual solenoidal realization.

Claimed quotient-space resolution

The preprint claims positive semigroup generation on the quotient by gradients, while proving that the usual solenoidal $L^1$ realization does not generate a $C_0$ -semigroup. If correct, this resolves the endpoint question by distinguishing the two realizations; the claim is unverified and has no independent confirmation in the retrieved evidence.

Current status (as of September 2026): A preprint claims a resolution, but its quotient-space generation result and non-generation result for the usual solenoidal $L^1$ realization remain unverified.

Sources

No solutions have been posted yet.