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Existence of Solutions of Nonconvex Multivalued Navier Stokes Equations

Bholanath Kumbhakar, Dwijendra Narain Pandey

TL;DR

This work addresses the existence of local strong solutions for the three-dimensional, nonstationary Navier–Stokes equations with a multivalued, nonconvex right-hand side. By constructing continuous selections through a decomposable-set framework and applying Schauder’s fixed-point theorem in a carefully defined Banach-space setting, the authors establish local solvability without requiring convex values of the multimap $F$. The results are first developed for the NSE with a multivalued forcing term and then extended to a general inclusion problem in an abstract operator setting, providing a robust approach to differential inclusions arising in fluid dynamics. The methods illuminate how nonconvex multivalued nonlinearities can be handled in high-dimensional evolution problems, with potential applications to viscoelastic and hysteretic models where set-valued right-hand sides naturally occur.

Abstract

In this paper, we discuss the existence of local strong solutions for the multivalued version of three-dimensional nonstationary Navier-Stokes equation in Banach spaces. Also, we considered a more general inclusion problem and studied the existence of solutions using the fixed point technique approach. We assume that the multivalued map possesses closed values (not necessarily convex values) and apply the Schauder Fixed Point Theorem in order to deduce the existence of fixed points.

Existence of Solutions of Nonconvex Multivalued Navier Stokes Equations

TL;DR

This work addresses the existence of local strong solutions for the three-dimensional, nonstationary Navier–Stokes equations with a multivalued, nonconvex right-hand side. By constructing continuous selections through a decomposable-set framework and applying Schauder’s fixed-point theorem in a carefully defined Banach-space setting, the authors establish local solvability without requiring convex values of the multimap . The results are first developed for the NSE with a multivalued forcing term and then extended to a general inclusion problem in an abstract operator setting, providing a robust approach to differential inclusions arising in fluid dynamics. The methods illuminate how nonconvex multivalued nonlinearities can be handled in high-dimensional evolution problems, with potential applications to viscoelastic and hysteretic models where set-valued right-hand sides naturally occur.

Abstract

In this paper, we discuss the existence of local strong solutions for the multivalued version of three-dimensional nonstationary Navier-Stokes equation in Banach spaces. Also, we considered a more general inclusion problem and studied the existence of solutions using the fixed point technique approach. We assume that the multivalued map possesses closed values (not necessarily convex values) and apply the Schauder Fixed Point Theorem in order to deduce the existence of fixed points.

Paper Structure

This paper contains 5 sections, 5 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multivalued Maps
  4. Navier Stokes Equations
  5. Main Results

Key Result

Lemma 2.1

MR1893419 Let $D\subset \mathbb{R}^N, (N\ge 2)$ be a bounded open set with $\partial D\in C^{2,\mu} (0<\mu<1)$ and assume $1<p,q<\infty, ~b>0$. If $u_0\in D_p^q, ~g\in L^q([0,b],H_p)$, then there exists a unique function $u$ satisfying $u\in L^q([0,b],D(A_p)), ~\partial_t u\in L^q([0,b],H_p)$ and Furthermore, where $c_A=c_A(\operatorname{Re}, p, q, D)$.

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 3.1
  • proof
  • Definition 4.1
  • ...and 3 more