Entropy Solution · Shun Li's Notes

The general form of the systems of conservation laws is

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where $\boldsymbol{u} = \boldsymbol{u}(\boldsymbol{x},t) \in \mathbb{R}^n$ , $\boldsymbol{f}(\boldsymbol{u}): \mathbb{R}^n \rightarrow \mathbb{R}^{d \times n}$ .

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where $j=1,\dots,n$ and $i=1,\dots,d$ .

Definition

A convex function $U(\boldsymbol{u}): \mathbb{R}^n \rightarrow \mathbb{R}$ is called an entropy function of $\eqref{eq:1}$ if there exists a function $\boldsymbol{F}(\boldsymbol{u}): \mathbb{R}^n \rightarrow \mathbb{R}^d$ , called the entropy flux, such that the following condition holds

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We can left-multiply $\partial U'(\boldsymbol{u})$ and obtian an extra conservation law for the entropy function

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Definition

A weak solution $\boldsymbol{u}$ of $\eqref{eq:1}$ is called an entropy solution if for all entropy functions $U$ , we have

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Integrating the entropy condition in space, we have

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That is, the total entropy is non-increasing with respect to time.

Define the entropy variables $\boldsymbol{v} = \boldsymbol{v}(\boldsymbol{u}) = \nabla U(\boldsymbol{u}) \in \mathbb{R}^n$ , i.e.

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The entropy condition can be written as

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If we assume that U is strictly convex, $U''(\boldsymbol{u}) > 0$ , the mapping $\boldsymbol{u} \to \boldsymbol{v}$ is one-to-one and can be regarded as a change of variables. Setting $\boldsymbol{u} = \boldsymbol{u}(\boldsymbol{v}): \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\boldsymbol{g}(\boldsymbol{v}) = \boldsymbol{f}(\boldsymbol{u}(\boldsymbol{v})): \mathbb{R}^n \rightarrow \mathbb{R}^{d \times n}$ , we can rewrite the system according to the entropy variables

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The following theorem tells us the symmetry of $g_i'(\boldsymbol{v})$ is equivalent to the existence of entropy function.

Theorem

A strictly convex function $U$ serves as an entropy function if and only if

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is symmetric positive-definite, and

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is symmetric for $i=1,\dots,d$ .

There exist functions $\boldsymbol{\phi}(\boldsymbol{v}): \mathbb{R}^n \rightarrow \mathbb{R}$ and $\boldsymbol{\psi}(\boldsymbol{v}): \mathbb{R}^n \rightarrow \mathbb{R}^d$ , called potential function and potential fluxes, such that

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The derivatives of $\phi(\boldsymbol{v})$ and $\psi^i(\boldsymbol{v})$ are as follows.

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The second-order derivatives of $\phi(\boldsymbol{v})$ and $\psi^i(\boldsymbol{v})$ are as follows.

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By left-multiplying the entropy variables to $\eqref{eq:pde1}$ and integrating by parts, one can obtain an entropy conservation law of the system on the continuous level satisfying

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The second term above is an entropy flux, measuring boundary effects on the total change in entropy inside the domain $\Omega$ .

Proof

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From the definitions of $\boldsymbol{F}$ and $\boldsymbol{\psi}$ , we obtain

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Thus,

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Examples (scalar)

Consider the conservation law

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Definition

A convex function $E(u)$ is called an entropy function of $\eqref{eq:conservation-law}$ if there exists a function $F(u)$ , called the entropy flux, such that

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By left-multiplying both sides of the equation by $E'(u)$ , we derive an additional conservation law satisfied by the entropy solution:

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Definition

A weak solution of $\eqref{eq:conservation-law}$ is called an entropy solution if, for all entropy pairs $(E,F)$ ,

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Define the entropy variable $\nu = \nu(u) = E'(u)$ . The entropy condition can be written as

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Assume that the mapping $u \to E'(u) = \nu$ is 1-1, which can be regarded as a change of variables. Let $u = u(\nu)$ and $g(\nu)=f(u(\nu))$ , we can rewrite the equation according to the entropy variable $\nu$ as

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Example

Consider the linear equation, $f(u) = c\,u$

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Solution

If we take $E(u) = \frac12 u^2$ and define $\nu = E'(u) = u$ , we have
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If we take $E(u) = e^u$ and define $\nu = E'(u) = e^u$ , we have
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Example

Consider the burgers equation, $f(u) = \frac{u^2}{2}$

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Solution

If we take $E(u) = \frac{u^2}2$ and define $\nu = E'(u) = u$ , we have
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If we take $E(u) = e^u$ and define $\nu = E'(u) = e^u$ , we have
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Example

Consider the Buckley-Leverett equation,

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Solution

If we take $E(u) = \frac{u^2}2$ and define $\nu = E'(u) = u$ , we have
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Examples: 1d Euler Equations

Example

Consider the 1d Euler equations, $\boldsymbol{u} = [\rho, \rho u, E]$

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Solution

The choice of entropy pair is

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where $s = \ln(\frac{p}{\rho^\gamma})$ . The entropy variable is

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