On the existence and uniqueness of solutions to coupled system of burgers equations

Introduction

Burger's equation is a simplified form of Navier Stokes Equation. In 1915, Harry Bateman [1] (1882-1946) modeled the initial and boundary value problem:

(1.1) (1.2) where and V are the velocity, spatial coordinate, time and kinematic viscosity respectively.

are prescribed functions of variables depending upon the specific conditions for the problem. The equation (1.1) is known as Burgers' equation acknowledging the contribution of Johannes Martinus Burgers [2] (1895-1981) who used it to describe the mathematical modeling of turbulence. Since the appearance of this problem, many researchers have worked on it in many fields of applications.

Formal Solution

In this paper we seek to prove the existence and uniqueness of solution of one dimensional coupled system of Burger's equation ( 1

( ,0) ( ), 0 (0, ) ( ), ( , ) ( ), 0

We set first part of our problem in the form of Theorem for the existence of solution.

If

for any , there exists classical solution to coupled system of Burgers equation (2.1)-(2.22). To prove this Theorem we shall require the following three steps:

1. Derivation of energy estimates. Here we determine upper bounds on norms of and their higher derivatives. This process is called a priori energy estimates, 2. Construction of approximate solutions. Here we seek the recourse of ODE theory to get local in time solution to a finite dimensional approximations of (2.1)-(2.2) satisfying same energy bounds as of the energy estimates, 3. Convergence of approximate solution to a solution. We use compactness argument to show that there exists a subsequence of that converges and that the limiting solution satisfies (2.1)-(2.2).

The following inequalities and lemma will be used in the derivation of a priori Energy estimates (a) Gronwall's Inequality [3] Assume for and it satisfies Then where (b) Young's inequality [4] (c) Cauchy Schwarz inequality [5] Lemma1: The following statements hold:

If and satisfies then

Proof 1. A Priori Energy Estimate

For a priori energy estimate we start of by taking the inner product of (2.1) (i) with

Integrating over by applying the divergence theorem and using the boundary conditions (2.2) on the boundary we have [5] (2.4) Then (2.4) becomes (2.5) Then using Cauchy Schwarz inequality and Young's inequality we have

Dropping the second term of the LHS and letting we get

Galerkin's approximation

The following definitions will be of essence when we discuss the Galerkin's approximation method Projection Operator: Let X be a vector space over K. A linear operator P on X is called a projection operator if

.The projection operator P satisfies the properties where denotes the inner product of square integrable functions.

Galerkin method: This is a method in which we approximate the solution of a PDE by the projection of the solution and the solution into finite dimensional subspaces (Hunter, 2014) [6] . Parsevals identity or equation: Let E be an inner product space. Then is a basis for E if and only if for arbitrary , , where

The identity above is called Parsevals identity or equation.

Using Galerkin approximation method we start off by constructing finite approximation to ( 1

Proposition 7 2: For any integer the solution found in Proposition (1) satisfies the same bounds as the a priori bounds on, for , , where is independent of and and only depends on bounds of . Furthermore , for , where is independent of and , but only depends on bounds of Proposition3 [7]: For any integer , the solution satisfies above Proposition1.

Proof

So, it follows that (2.50)

Thus we have shown that the finite dimensional approximation of (2.1)-(2.2) satisfies same energy bounds as of a priori energy estimate. [8] We shall use following Lemmas and Arzela's dominated convergence theorem for the proof of convergence of approximate solution to a solution of the system of Burgers Equation ( 2 Furthermore, using previous lemma, for any given , subsequences converges uniformly for as

Convergence of approximate solution