Existence and Uniqueness of the Viscous Burgers’ Equation with the p-Laplace Operator

1. Introduction

The Burgers’ equation, first introduced by J.M. Burgers, is a fundamental model for various physical processes, including shock wave propagation and turbulence [1]. Originally formulated for Newtonian fluids, the Burgers’ equation has been extensively analyzed; however, its generalization to non-Newtonian fluids—specifically those described by the power-law model—has received comparatively less attention for the case when $1<p<2$ [2]. We employ a particular case of the generalized Burgers’ equation formulated by Wei and Jordan [3], which was used to analyze acoustic propagation in power-law fluids. Traveling wave solutions of this equation were derived by Wei and Borden [4].

This paper’s main focus is to investigate the existence and uniqueness of weak solutions to the generalized viscous Burgers’ equation. The study is framed within the context of Sobolev spaces, which provide a robust mathematical environment for addressing weak solutions and ensuring appropriate regularity conditions [5].

Establishing the existence of weak solutions to nonlinear partial differential equations (PDEs) like the generalized Burgers’ equation has been well-studied in the context of Newtonian fluids [6,7]. However, extending these results to the power-law non-Newtonian fluids involves additional complexities due to the nonlinearity introduced by the fluid’s viscosity [8]. The unique solutions for the generalized Burgers’ equation based on power-law fluids remain largely unexplored for the case when $p\ne 2$, creating a gap in the current literature that this paper aims to address.

Wei and Jordan [3] established the following Burgers’ equation

$$\rho ({\partial }_{t}u+u{\partial }_{x}u)=-P_x+{\partial }_x\left[\left({\mu }_{\mathrm{B}}+{\displaystyle \frac{4}{3}}\mu k{\left|{\partial }_{x}u\right|}^{m-1}\right){\partial }_{x}u\right]$$

as a model to study acoustic traveling waves in Power-law fluids with

$m=p-1$, $0<m<1$ is the power-law index.

In this paper, we study the well-posedness and numerical solutions of a special case of the above equation with initial and boundary conditions

$$\left\{\begin{array}{c}\rho ({\partial }_{t}u+u{\partial }_{x}u)=\mu {\partial }_x\left({\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u\right),\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in [0,L],\hfill \\ u(t,0)=u(t,L)=0,\hfill & t\in [0,T].\hfill \end{array}\right.$$

By using the dimensionless variables $x^{*}={\displaystyle \frac{1}{L}}x$, $u^{*}={\displaystyle \frac{1}{u_0}}u$, $t^{*}={\displaystyle \frac{u_0}{L}}t$, to the last equation, we obtain the following problem, in which $x^{*}$, $u^{*}$, $t^{*}$ are still denoted by x, u, t

$$\left\{\begin{array}{c}{\partial }_{t}u+u{\partial }_{x}u={\displaystyle \frac{1}{Re}}{\partial }_x\left({\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u\right),\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in [0,1],\hfill \\ u(t,0)=u(t,1)=0,\hfill & t\in [0,T]\hfill \end{array}\right.$$

(1)

with $1<p<2$ and $m=p-1$. In equation (1), $Re$ stands for the Reynolds number, a dimensionless quantity determining the flow behavior based on the balance of inertial and viscous forces. The Reynolds number for power-law fluids is given by

$$Re={\displaystyle \frac{\rho L^{p-1}}{\mu u_0^{p-3}}},$$

where L and $u_0$ represent the characteristic length and velocity, respectively (see, for example, [9]). The Reynolds number significantly influences the stability of the flow and the type of flow regime observed. Our numerical simulations demonstrate this. When in case $p=2$, equation (1) becomes a well-known Burgers’ equation for a Newtonian fluid

$$\left\{\begin{array}{c}{\partial }_{t}u+u{\partial }_{x}u={\displaystyle \frac{1}{Re}}{\partial }_x^{2}u,\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in [0,1],\hfill \\ u(t,0)=u(t,1)=0,\hfill & t\in [0,T].\hfill \end{array}\right.$$

The well-posedness of the viscous Burgers’ equation for Newtonian fluid has been studied intensively in the literature (see, for example, [6,10,11]). To our knowledge, the well-posedness of the problem (1) has not been studied before. In our work, we prove the result concerning the existence, uniqueness, and regularity of a solution to the Burgers equation with p-Laplasian right-hand side (1).

Multiplying the equation of (1) by a test function $w\in W_0^{1,p}(0,1)$, and integrating by parts from 0 to 1, we convert the initial boundary value problem (1) to the following integral equation

$${\int }_0^1{\partial }_{t}uwdx+{\int }_0^{1}u{\partial }_{x}uwdx+{\displaystyle \frac{1}{Re}}{\int }_0^1\left({\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u\right){\partial }_{x}wdx=0$$

(2)

for all $w\in W_0^{1,p}(0,1)$, $t\in (0,T)$ and $1<p<2$.

Definition 1.1. 

For $1<p<2$, a function $u\in L^2\left(0,T;{W_0}^{1,p}(0,1)\right)$, for which ${\partial }_{t}u\in$$L^2\left(0,T;L^2(0,1)\right)$ is said to be a weak solution of problem (1) if it satisfies equation (2) for each $w\in {W_0}^{1,p}(0,1)$ and a.e. $t\in (0,T)$ and $u(0,x)=u_0$ for all $x\in (0,1)$.

Now, we present the main result of the work.

Theorem 1.2. 

Let $u_0\in W_0^{1,p}(0,1)$. Then there exists a unique solution $u\in L^2\left(0,T;{W_0}^{1,p}(0,1)\right)$ of (1) such that ${\partial }_{t}u\in L^2\left(0,T;L^2(0,1)\right)$, in the sense of Definition 1.1.

In Section 2, we state some preliminary information needed to study the well-posedness of the problem (1), formulated in the weak form (2). Section 3 consists of two subSection 3.1 and Section 3.2. We obtain the existence result in Section 3.1 and establish the uniqueness of the solution in Section 3.2. Moreover, in Section 4, we construct the numerical solutions for the various values of the Reynolds number $Re$, by using the COMSOL PDE solver with some numerical manipulations to avoid singularity in ${▵}_{p}u$ at points where ${\partial }_{x}u=0$.

2. Preliminary

We present well-known facts that we will use to prove Theorem 1.2. The Gronwall’s inequality, some Sobolev inequalities, and the monotonicity of the p-Laplacian operator defined by ${▵}_{p}u={\partial }_x\left({\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u\right)$ are required to establish the main result.

Recall that $L^p(0,1)$ and $W^{k,p}(0,1)$ are the standard spaces of Lebesgue and Sobolev, respectively, for $1<p<\infty$ and $k\in \mathbb{Z}$. For any Banach space X, we define $L^p(0,T;X)$ to be the space of measurable functions $u:(0,T)\to X$ such that

$${\parallel u\parallel }_{L^p(0,T;X)}={\left({\int }_0^T{\parallel u\parallel }_X^p\mathrm{d}t\right)}^{1/p}<\infty$$

for $1<p<\infty$ and ${\parallel u\parallel }_{L^{\infty }(0,T;X)}=ess{sup}_{0<t<T}{\parallel u\parallel }_X<\infty$ if $p=\infty$. $L^p(0,T;X)$ is a Banach space [12].

Proposition 2.1 

(The differential form of Gronwall’s inequality [13]).

(i)

Let $\eta (·)$ be a nonnegative, absolutely continuous function on $[0,T]$, which satisfies for a.e. t the differential inequality

$${\eta }^{\prime }\left(t\right)\le \varphi \left(t\right)\eta \left(t\right)+\psi \left(t\right),$$

where $\varphi \left(t\right)$ and $\psi \left(t\right)$ are nonnegative, summable functions on $[0,T]$. Then

$$\eta \left(t\right)\le e^{{\int }_0^t\varphi \left(s\right)ds}\left[\eta \left(0\right)+{\int }_0^t\psi \left(s\right)ds\right]$$

for all $0\le t\le T$.

(ii)

In particular, if

$${\eta }^{\prime }\le \varphi \eta \mathit{on}[0,T]\mathit{and}\eta \left(0\right)=0$$

then

$$\eta \equiv 0\mathit{on}[0,T].$$

Proposition 2.2 (Poincaré’s inequality

[13]). Assume U is a bounded, open subset of ${\mathbb{R}}^n$. Suppose $u\in W_0^{1,p}\left(U\right)$ for some $1\le p<n$. Then we have the estimate

$${\parallel u\parallel }_{L^q\left(U\right)}\le C{\parallel Du\parallel }_{L^p\left(U\right)}$$

for each $q\in [1,{\displaystyle \frac{np}{n-p}}]$, the constant C depending only on $p,q,n$ and U.

In particular, for all $1\le p\le \infty$,

$${\parallel u\parallel }_{L^p\left(U\right)}\le C{\parallel Du\parallel }_{L^p\left(U\right)}.$$

(3)

Next, we recall the Bounded Convergence Theorem.

Proposition 2.3 

([14]). If f is a Lebesgue measurable function defined on a closed, bounded interval [a,b], with its $L^p$ norm for $1<p<\infty$ defined by

$${\parallel f\parallel }_p={\left({\int }_a^b{\left|f\left(x\right)\right|}^{p}dx\right)}^{1/p}$$

and $L^{\infty }$ norm

$${\parallel f\parallel }_{\infty }=\mathrm{ess}.\sup .\left\{\right|f\left(x\right)|:a\leqq x\leqq b\}.$$

If $f_1,f_2,f_3,\cdots$ is a sequence of measurable functions and M is a positive constant such that

$${∥f_n∥}_{\infty }\leqq Mn=1,2,\cdots$$

and ${lim}_{n\to \infty }{f}_n\left(x\right)=f\left(x\right)$ a.e. $x\in [a,b]$ then

$$\underset{n\to \infty }{lim}{∥f_n-f∥}_p=0$$

for any p satisfying $0<p<\infty$.

Now, we state the well-known Theorem.

Proposition 2.4 

([15]). The space $L^2(0,T;W_0^{1,p}(0,1))$ is complete and separable if $1<p<\infty$.

Proposition 2.5 (Jensen’s inequality)

[16] Let $(\Omega ,\Lambda ,\mu )$ be a measure space with $0<\mu \left(\Omega \right)<\infty$ and let $\varphi :I\to \mathbb{R}$ be a convex function defined on an open interval I in $\mathbb{R}$ if $f:\Omega \to I$ is such that $f,\varphi \circ f\in L(\Omega ,\Lambda ,\mu )$, then

$$\varphi \left({\displaystyle \frac{1}{\mu \left(\Omega \right)}}{\int }_{\Omega }fd\mu \right)\le {\displaystyle \frac{1}{\mu \left(\Omega \right)}}{\int }_{\Omega }\varphi \left(f\right)d\mu .$$

Proposition 2.6 

([17]). If a sequence $\left(f_n\right)$ of continuous functions $f_n:A\to \mathbb{R}$ converges uniformly on $A\subset \mathbb{R}$ to $f:A\to \mathbb{R}$, then f is continuous on A.