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Existence and approximate controllability results for time-fractional stochastic Navier-Stokes equations

Renu Chaudhary, Simeon Reich, Juan J. Nieto

TL;DR

This paper advances the theory of time-fractional stochastic Navier-Stokes equations by proving the existence and uniqueness of mild solutions under a Caputo derivative of order $\eta$ and a fractional Laplacian of order $\alpha$, in a bounded domain. It then establishes approximate controllability of the system by constructing a fixed-point framework with generalized Mittag-Leffler operators and analyzing the stochastic Gramian, showing that controllability can be achieved in the sense that the reachable set is dense in $L^p(\Sigma,H)$. A key contribution is coupling memory effects and randomness within a rigorous controllability analysis, using a Banach contraction argument and control-operator techniques. An explicit 2D numerical example with Grünwald–Letnikov time discretization and spectral spatial discretization corroborates the theoretical results, illustrating steering toward a target Fourier mode under stochastic perturbations.

Abstract

This paper deals with time-fractional stochastic Navier-Stokes equations, which are characterized by the coexistence of stochastic noise and a fractional power of the Laplacian. We establish sufficient conditions for the existence and approximate controllability of a unique mild solution to time-fractional stochastic Navier-Stokes equations. Using a fixed point technique, we first demonstrate the existence and uniqueness of a mild solution to the equation under consideration. We then establish approximate controllability results by using the concepts of fractional calculus, semigroup theory, functional analysis and stochastic analysis.

Existence and approximate controllability results for time-fractional stochastic Navier-Stokes equations

TL;DR

This paper advances the theory of time-fractional stochastic Navier-Stokes equations by proving the existence and uniqueness of mild solutions under a Caputo derivative of order and a fractional Laplacian of order , in a bounded domain. It then establishes approximate controllability of the system by constructing a fixed-point framework with generalized Mittag-Leffler operators and analyzing the stochastic Gramian, showing that controllability can be achieved in the sense that the reachable set is dense in . A key contribution is coupling memory effects and randomness within a rigorous controllability analysis, using a Banach contraction argument and control-operator techniques. An explicit 2D numerical example with Grünwald–Letnikov time discretization and spectral spatial discretization corroborates the theoretical results, illustrating steering toward a target Fourier mode under stochastic perturbations.

Abstract

This paper deals with time-fractional stochastic Navier-Stokes equations, which are characterized by the coexistence of stochastic noise and a fractional power of the Laplacian. We establish sufficient conditions for the existence and approximate controllability of a unique mild solution to time-fractional stochastic Navier-Stokes equations. Using a fixed point technique, we first demonstrate the existence and uniqueness of a mild solution to the equation under consideration. We then establish approximate controllability results by using the concepts of fractional calculus, semigroup theory, functional analysis and stochastic analysis.
Paper Structure (6 sections, 9 theorems, 79 equations, 2 figures)

This paper contains 6 sections, 9 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Existence and Controllability Results
  4. Example
  5. Numerical Scheme
  6. Results and Interpretation

Key Result

Lemma 1

zou For any $\alpha>0$, the operator $-A_\alpha$ generates an analytic semigroup $S_\alpha(t)=e^{-tA_\alpha}$, $t\geq 0$, on $L^p$. Moreover, we have where $\beta\geq0$, the constant $C_{\alpha,\beta}>0$ depends on $\alpha$ and $\beta$, and $\mathcal{L}(L^p)$ denotes the Banach space of all bounded linear operators from $L^p$ into itself.

Figures (2)

  • Figure 1: Logical structure of the paper: Lemmas, Definitions, Assumptions, and Theorems with explicit dependencies.
  • Figure 2: Time evolution of the $L^2$ norm of the velocity field $\|z(t,\cdot,\cdot)\|_{L^2}$ under the controlled and uncontrolled stochastic fractional Navier–Stokes dynamics.

Theorems & Definitions (17)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Definition 2
  • Lemma 3
  • Lemma 4
  • Definition 3
  • Lemma 5
  • Lemma 6
  • proof
  • ...and 7 more