Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and Uniqueness of viscosity solutions of Value function of Local Cahn-Hilliard-Navier-Stokes system

Sheetal Dharmatti, Perisetti Lakshmi Naga Mahendranath

Abstract

In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in two dimensional bounded domain. We formulate distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier- Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via comparison principle.

Existence and Uniqueness of viscosity solutions of Value function of Local Cahn-Hilliard-Navier-Stokes system

Abstract

In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in two dimensional bounded domain. We formulate distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier- Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via comparison principle.

Paper Structure

This paper contains 7 sections, 17 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Mathematical setting
  3. Dynamic Programming Principle
  4. Continuity properties of a value function
  5. Dynamic Programming Principle (DPP)
  6. Hamilton Jacobi Bellman equations
  7. Uniqueness

Key Result

Lemma 1

Let $\Omega\subset\mathbb{R}^n$, $\mathbf{u}\in\mathbb{W}^{m,p}(\Omega;\mathbb{R}^n), p\geq 1$ and fix $1 \leq p,q \leq \infty$ and a natural number $m$. Suppose also that a real number $\theta$ and a natural number $j$ are such that and $\frac{j}{m} \leq \theta \leq 1.$ Then for any $\mathbf{u}\in\mathbb{W}^{m,p}(\Omega;\mathbb{R}^n),$ we have where $s > 0$ is arbitrary and the constant $C$ dep

Theorems & Definitions (31)

  • Lemma 1: Gagliardo-Nirenberg interpolation inequality, Theorem 1, MR109940
  • Lemma 2: Ladyzenskaya's inequality
  • Lemma 3: Agmon's inequality
  • Lemma 4: Ponicare-Wirtinger inequality
  • Lemma 5: MR1918929
  • Definition 1
  • Definition 2
  • Theorem 2.1
  • Theorem 2.2: Proposition 2.2, MR3436705
  • Theorem 2.3
  • ...and 21 more