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On the existence of approximate problems that preserve the type of a bifurcation point of a nonlinear problem. Application to the stationary Navier-Stokes equations

Cătălin Liviu Bichir

TL;DR

The paper develops a general framework to ensure that an approximate reformulation of a steady-state bifurcation problem on Banach spaces preserves the bifurcation type of the exact problem. By embedding the original problem into an extended, overdetermined system and employing Graves' theorem and metric-regularity concepts for set-valued maps, the authors construct an approximate equation of the form $F_{h}(\lambda_{h},u_{h})-\varrho_{h}=0$ with a bifurcation point of the same type as the exact one. They establish conditions under which this extended-system approach yields a local $C^{p}$-equivalence to a perturbed version of the original map, and they apply the theory to the Dirichlet problem for the stationary Navier–Stokes equations, detailing the finite-element-like approximation framework and verification steps. The results provide a robust route to study qualitative bifurcation features of nonlinear problems via accurate and controllable approximations, with implications for nonlinear functional analysis, singularity theory, and hydrodynamic stability computations. Overall, the work offers a principled method to reconcile infinite-dimensional bifurcation theory with practical numerical approximations, preserving essential bifurcation characteristics in the approximate problems.

Abstract

We consider a nonlinear problem $F(λ,u)=0$ on infinite-dimensional Banach spaces that correspond to the steady-state bifurcation case. In the literature, it is found again a bifurcation point of the approximate problem $F_{h}(λ_{h},u_{h})=0$ only in some cases. We prove that, in every situation, given $F_{h}$ that approximates $F$, there exists an approximate problem $F_{h}(λ_{h},u_{h})-\varrho_{h} = 0$ that has a bifurcation point with the same properties as the bifurcation point of $F(λ,u)=0$. First, we formulate, for a function $\widehat{F}$ defined on general Banach spaces, some sufficient conditions for the existence of an equation that has a bifurcation point of certain type. For the proof of this result, we use some methods from variational analysis, Graves' theorem, one of its consequences and the contraction mapping principle for set-valued mappings. These techniques allow us to prove the existence of a solution with some desired components that equal zero of an overdetermined extended system. We then obtain the existence of a constant (or a function) $\widehat{\varrho}$ so that the equation $\widehat{F}(λ,u)-\widehat{\varrho} = 0$ has a bifurcation point of certain type. This equation has $\widehat{F}(λ,u) = 0$ as a perturbation. It is also made evident a class of maps $C^{p}$ - equivalent (right equivalent) at the bifurcation point to $\widehat{F}(λ,u)-\widehat{\varrho}$ at the bifurcation point. Then, for the study of the approximation of $F(λ,u)=0$, we give conditions that relate the exact and the approximate functions. As an application of the theorem on general Banach spaces, we formulate conditions in order to obtain the existence of the approximate equation $F_{h}(λ_{h},u_{h})-\varrho_{h} = 0$.

On the existence of approximate problems that preserve the type of a bifurcation point of a nonlinear problem. Application to the stationary Navier-Stokes equations

TL;DR

The paper develops a general framework to ensure that an approximate reformulation of a steady-state bifurcation problem on Banach spaces preserves the bifurcation type of the exact problem. By embedding the original problem into an extended, overdetermined system and employing Graves' theorem and metric-regularity concepts for set-valued maps, the authors construct an approximate equation of the form with a bifurcation point of the same type as the exact one. They establish conditions under which this extended-system approach yields a local -equivalence to a perturbed version of the original map, and they apply the theory to the Dirichlet problem for the stationary Navier–Stokes equations, detailing the finite-element-like approximation framework and verification steps. The results provide a robust route to study qualitative bifurcation features of nonlinear problems via accurate and controllable approximations, with implications for nonlinear functional analysis, singularity theory, and hydrodynamic stability computations. Overall, the work offers a principled method to reconcile infinite-dimensional bifurcation theory with practical numerical approximations, preserving essential bifurcation characteristics in the approximate problems.

Abstract

We consider a nonlinear problem on infinite-dimensional Banach spaces that correspond to the steady-state bifurcation case. In the literature, it is found again a bifurcation point of the approximate problem only in some cases. We prove that, in every situation, given that approximates , there exists an approximate problem that has a bifurcation point with the same properties as the bifurcation point of . First, we formulate, for a function defined on general Banach spaces, some sufficient conditions for the existence of an equation that has a bifurcation point of certain type. For the proof of this result, we use some methods from variational analysis, Graves' theorem, one of its consequences and the contraction mapping principle for set-valued mappings. These techniques allow us to prove the existence of a solution with some desired components that equal zero of an overdetermined extended system. We then obtain the existence of a constant (or a function) so that the equation has a bifurcation point of certain type. This equation has as a perturbation. It is also made evident a class of maps - equivalent (right equivalent) at the bifurcation point to at the bifurcation point. Then, for the study of the approximation of , we give conditions that relate the exact and the approximate functions. As an application of the theorem on general Banach spaces, we formulate conditions in order to obtain the existence of the approximate equation .
Paper Structure (32 sections, 44 theorems, 238 equations)

This paper contains 32 sections, 44 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The setting of Crouzeix and Rappaz CLBichir_bib_Cr_Ra1990
  4. Contraction mapping principle for set-valued mappings and Graves' theorem CLBichir_bib_Dontchev_Rockafellar2009
  5. The main theorem on the extended systems considered in the work
  6. Proof of the main theorem on the extended systems, Theorem \ref{['teorema_principala_parteaMAIN']}
  7. Some properties of the extended systems associated to $D_{u}F(\lambda,u)$ and $DF(\lambda,u)$
  8. The properties of kernel of $DF(\lambda_{0},u_{0})$ and of $D_{u}F(\lambda_{0},u_{0})$
  9. The decompositions of space $Z$
  10. Properties of the Fréchet derivatives and the consequences on the qualitative aspects
  11. The proof of implication (i) $\Rightarrow$ (ii) of Theorem \ref{['teorema_principala_parteaMAIN']}
  12. The proof of implication (ii) $\Rightarrow$ (i) of Theorem \ref{['teorema_principala_parteaMAIN']}
  13. The main result on the existence of a bifurcation problem
  14. Introduction
  15. The definition of the mappings $\mathcal{G}(s,\phi')$ and $\mathcal{Q}(s,\phi')$
  16. ...and 17 more sections

Key Result

Lemma 2.1

(Lemma 4.1 and Lemma 6.1, CLBichir_bib_Cr_Ra1990). Under the above hypotheses, there exists a neighborhood $\mathcal{V}$ of $0$ in $\mathbb{R}^{q+m}$, a neighborhood $\mathcal{U}$ of $x_{0}$ in $X$ and a unique $C^{p}$ - mapping $x : \theta \in \mathcal{V} \rightarrow x(\theta)=(f(\theta),\lambda(\t

Theorems & Definitions (77)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 67 more