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Local-in-time strong solvability of Navier--Stokes type variational inequalities by Rothe's method

Takahito Kashiwabara

TL;DR

This work develops a Rothe-time discretization framework to establish local-in-time strong solvability for Navier–Stokes type variational inequalities in a Hilbert space setting, without requiring the cancelation property $\langle B(u,v),v\rangle=0$ and under a stationarity-based regularity assumption on the Stokes problem. By combining a semi-implicit time scheme with a fully implicit monotone term $\varphi$, the authors derive uniform a priori estimates and prove existence and uniqueness of a local strong solution in the maximal-$L^2$ regularity class, and, under stronger data, in the Kiselev–Ladyzhenskaya class. A solvability theory for stationary and Oseen-type problems supports the time-discrete scheme, and the approach is illustrated through four concrete models: obstacle problems for convection–diffusion, nonlinear Neumann problems, Navier–Stokes with friction-type boundary conditions, and a Bingham flow with slip. The results extend the applicability of NS-type variational inequalities to broader boundary conditions and nonlinearities, providing a constructive, time-discretization-based pathway to local-in-time regular solutions in three dimensions. Overall, the paper advances a rigorous framework for local strong solvability of parabolic VIs with NS-type nonlinearities using Rothe’s method and stationary-stokes regularity assumptions.

Abstract

We consider parabolic variational inequalities in a Hilbert space $V$, which have a non-monotone nonlinearity of Navier--Stokes type represented by a bilinear operator $B: V \times V \to V'$ and a monotone type nonlinearity described by a convex, proper, and lower-semicontinuous functional $\varphi : V \to (-\infty, +\infty]$. Existence and uniqueness of a local-in-time strong solution in a maximal-$L^2$-regularity class and in a Kiselev--Ladyzhenskaya class are proved by discretization in time (also known as Rothe's method), provided that a corresponding stationary Stokes problem admits a regularity structure better than $V$ (which is typically $H^2$-regularity in case of the Navier--Stokes equations). Since we do not assume the cancelation property $\left< B(u, v), v \right> = 0$, in applications we may allow for broader boundary conditions than those treated by the existing literature.

Local-in-time strong solvability of Navier--Stokes type variational inequalities by Rothe's method

TL;DR

This work develops a Rothe-time discretization framework to establish local-in-time strong solvability for Navier–Stokes type variational inequalities in a Hilbert space setting, without requiring the cancelation property and under a stationarity-based regularity assumption on the Stokes problem. By combining a semi-implicit time scheme with a fully implicit monotone term , the authors derive uniform a priori estimates and prove existence and uniqueness of a local strong solution in the maximal- regularity class, and, under stronger data, in the Kiselev–Ladyzhenskaya class. A solvability theory for stationary and Oseen-type problems supports the time-discrete scheme, and the approach is illustrated through four concrete models: obstacle problems for convection–diffusion, nonlinear Neumann problems, Navier–Stokes with friction-type boundary conditions, and a Bingham flow with slip. The results extend the applicability of NS-type variational inequalities to broader boundary conditions and nonlinearities, providing a constructive, time-discretization-based pathway to local-in-time regular solutions in three dimensions. Overall, the paper advances a rigorous framework for local strong solvability of parabolic VIs with NS-type nonlinearities using Rothe’s method and stationary-stokes regularity assumptions.

Abstract

We consider parabolic variational inequalities in a Hilbert space , which have a non-monotone nonlinearity of Navier--Stokes type represented by a bilinear operator and a monotone type nonlinearity described by a convex, proper, and lower-semicontinuous functional . Existence and uniqueness of a local-in-time strong solution in a maximal--regularity class and in a Kiselev--Ladyzhenskaya class are proved by discretization in time (also known as Rothe's method), provided that a corresponding stationary Stokes problem admits a regularity structure better than (which is typically -regularity in case of the Navier--Stokes equations). Since we do not assume the cancelation property , in applications we may allow for broader boundary conditions than those treated by the existing literature.
Paper Structure (12 sections, 13 theorems, 99 equations)

This paper contains 12 sections, 13 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. Obstacle problem for convection-diffusion equations
  5. Nonlinear Neumann problems for convection-diffusion equations
  6. Navier--Stokes equations with boundary conditions of friction type
  7. Bingham problem with perfect slip boundary condition
  8. Solvability of stationary problems
  9. Discrete-in-time approximation and a priori estimates
  10. Passage to limit $\Delta t \to 0$ and proof of Theorem \ref{['main thm1']}
  11. Proof of Theorem \ref{['main thm2']}
  12. Remarks on solution in Kiselev--Ladyzhenskaya class

Key Result

Theorem 2.1

Under the hypotheses of (H1)--(H5) above, let $f \in L^2(0, T; H)$ and $u^0 \in D(\varphi) \subset V$ be arbitrary. Then there exist some $T_* \in (0, T]$ and a unique $u \in H^1(0, T_*; H) \cap L^\infty(0, T_*; V) \cap L^2(0, T_*; W)$ that solves (eq: VI) for a.e. $t \in (0, T)$.

Theorems & Definitions (32)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 22 more