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On a mixed problem for the nonstationary Stokes system in an angle

Jürgen Rossmann

TL;DR

The paper analyzes a time-dependent Stokes system in a planar angle with mixed Dirichlet and Neumann boundary data. By applying the Laplace transform, it reduces to a parameter-dependent stationary problem and develops a robust framework in weighted Sobolev spaces to establish existence, uniqueness, and regularity of both weak and strong solutions. Central to the analysis are the operator pencils governing corner singularities, normal solvability criteria, and regularity results that ensure well-posedness across a range of weights. Finally, these results are extended back to the time domain to obtain unique, time-dependent solutions with explicit a priori estimates, illustrating the approach's effectiveness for PDEs in non-smooth domains with mixed boundary conditions.

Abstract

The autor considers an initial-boundary value problem for the nonstationary Stokes system in an angle, where Dirichlet and Neumann conditions are prescribed on the diferent sides of the angle. The major part of the paper deals with the parameter-depending problem which arises after the application of the Laplace transform. The author obtains existence, uniqueness and regularity results for this problem in a class of weighted Sobolev spaces. Using the properties of the Laplace transform, he proves the existence and uniqueness of strong solutions of the time-dependent problem.

On a mixed problem for the nonstationary Stokes system in an angle

TL;DR

The paper analyzes a time-dependent Stokes system in a planar angle with mixed Dirichlet and Neumann boundary data. By applying the Laplace transform, it reduces to a parameter-dependent stationary problem and develops a robust framework in weighted Sobolev spaces to establish existence, uniqueness, and regularity of both weak and strong solutions. Central to the analysis are the operator pencils governing corner singularities, normal solvability criteria, and regularity results that ensure well-posedness across a range of weights. Finally, these results are extended back to the time domain to obtain unique, time-dependent solutions with explicit a priori estimates, illustrating the approach's effectiveness for PDEs in non-smooth domains with mixed boundary conditions.

Abstract

The autor considers an initial-boundary value problem for the nonstationary Stokes system in an angle, where Dirichlet and Neumann conditions are prescribed on the diferent sides of the angle. The major part of the paper deals with the parameter-depending problem which arises after the application of the Laplace transform. The author obtains existence, uniqueness and regularity results for this problem in a class of weighted Sobolev spaces. Using the properties of the Laplace transform, he proves the existence and uniqueness of strong solutions of the time-dependent problem.

Paper Structure

This paper contains 14 sections, 15 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. The mixed problem for the stationary Stokes system
  3. Weak solutions of the parameter-depending problem in the angle $K$
  4. Weighted spaces in an angle
  5. Solutions of the equation $\hbox{div}\, u = g$
  6. Existence of weak solutions
  7. Strong solutions of the parameter-dependent problem
  8. An estimate for the function $p$
  9. Normal solvability
  10. A regularity assertion for strong solutions
  11. Existence of strong solutions
  12. The time-dependent problem in $K$
  13. Weighted Sobolev spaces in $K\times{\Bbb R}_+$
  14. Existence and uniqueness of solutions

Key Result

Theorem 2.1

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • Corollary 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Corollary 4.1
  • Lemma 4.4
  • ...and 5 more