Convexity and entropy for conservation laws

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I am reading Hyperbolic Systems of Conservation Laws by Godlewski and Raviart (1).

1. While defining the entropy why do we consider only convex flux?
   Can it be extended to general entropy flux? Please suggest me the reference.


2. I understood that entropy conditions are the conditions which assures the uniqueness of the solution. Why do we call them as entropy condition? Is it related to the entropy which we study in Thermodynamics?

---

(1) E. Godlewski, P.-A. Raviart, Hyperbolic Systems of Conservation Laws. Ellipses, 1991.

pde hyperbolic-equations

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edited Aug 9 at 6:28

Harry49

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up vote
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I am reading Hyperbolic Systems of Conservation Laws by Godlewski and Raviart (1).

1. While defining the entropy why do we consider only convex flux?
   Can it be extended to general entropy flux? Please suggest me the reference.


2. I understood that entropy conditions are the conditions which assures the uniqueness of the solution. Why do we call them as entropy condition? Is it related to the entropy which we study in Thermodynamics?

---

(1) E. Godlewski, P.-A. Raviart, Hyperbolic Systems of Conservation Laws. Ellipses, 1991.

pde hyperbolic-equations

share|cite|improve this question

edited Aug 9 at 6:28

Harry49

4,6661825

asked Jul 26 at 6:11

Rosy

643

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

I am reading Hyperbolic Systems of Conservation Laws by Godlewski and Raviart (1).

1. While defining the entropy why do we consider only convex flux?
   Can it be extended to general entropy flux? Please suggest me the reference.


2. I understood that entropy conditions are the conditions which assures the uniqueness of the solution. Why do we call them as entropy condition? Is it related to the entropy which we study in Thermodynamics?

---

(1) E. Godlewski, P.-A. Raviart, Hyperbolic Systems of Conservation Laws. Ellipses, 1991.

pde hyperbolic-equations

share|cite|improve this question

edited Aug 9 at 6:28

Harry49

4,6661825

asked Jul 26 at 6:11

Rosy

643

I am reading Hyperbolic Systems of Conservation Laws by Godlewski and Raviart (1).

1. While defining the entropy why do we consider only convex flux?
   Can it be extended to general entropy flux? Please suggest me the reference.


2. I understood that entropy conditions are the conditions which assures the uniqueness of the solution. Why do we call them as entropy condition? Is it related to the entropy which we study in Thermodynamics?

---

(1) E. Godlewski, P.-A. Raviart, Hyperbolic Systems of Conservation Laws. Ellipses, 1991.

pde hyperbolic-equations

share|cite|improve this question

edited Aug 9 at 6:28

Harry49

4,6661825

asked Jul 26 at 6:11

Rosy

643

share|cite|improve this question

share|cite|improve this question

edited Aug 9 at 6:28

Harry49

4,6661825

edited Aug 9 at 6:28

Harry49

4,6661825

edited Aug 9 at 6:28

Harry49

4,6661825

4,6661825

asked Jul 26 at 6:11

Rosy

643

asked Jul 26 at 6:11

Rosy

643

asked Jul 26 at 6:11

Rosy

643

643

add a comment | 

add a comment | 

1 Answer
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Let us consider the scalar one-dimensional case $u_t + f(u)_x = 0$ for sake of simplicity, as described in Section 3.8 of (1).

An entropy function $eta$ is a convex function of $u$ which satisfies the conservation law $eta(u)_t + psi(u)_x = 0$, where $psi$ is the entropy flux. In the definition, neither the flux $f$ nor the entropy flux $psi$ is convex, but only $eta$ is convex. This is a conventional choice with important consequences, "though one may consider a more general
framework" (2). If $eta$ is chosen convex, then the vanishing viscosity weak solution -- i.e., the physically relevant weak solution -- makes the entropy decrease in the weak sense. This leads to the definition of the entropy solution, which is a weak solution $u$ such that $eta(u)_t + psi(u)_x leq 0$ weakly for all convex entropy function $eta$.

As such, the definition of an entropy function $eta$ is not related to the thermodynamical notion of entropy. Firstly, it decreases over time, while the entropy increases in thermodynamics. However, there may be a link with the thermodynamical entropy for some particular systems. In the case of Eulerian gas dynamics, a natural mathematical entropy function is defined with respect to the thermodynamical entropy (cf. Section II.2 of (2)).

---

(1) R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1

(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996. doi:10.1007/978-1-4612-0713-9

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edited Aug 9 at 6:30

answered Aug 8 at 14:38

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1 Answer
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1 Answer
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oldest

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oldest

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oldest

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up vote
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down vote

Let us consider the scalar one-dimensional case $u_t + f(u)_x = 0$ for sake of simplicity, as described in Section 3.8 of (1).

An entropy function $eta$ is a convex function of $u$ which satisfies the conservation law $eta(u)_t + psi(u)_x = 0$, where $psi$ is the entropy flux. In the definition, neither the flux $f$ nor the entropy flux $psi$ is convex, but only $eta$ is convex. This is a conventional choice with important consequences, "though one may consider a more general
framework" (2). If $eta$ is chosen convex, then the vanishing viscosity weak solution -- i.e., the physically relevant weak solution -- makes the entropy decrease in the weak sense. This leads to the definition of the entropy solution, which is a weak solution $u$ such that $eta(u)_t + psi(u)_x leq 0$ weakly for all convex entropy function $eta$.

As such, the definition of an entropy function $eta$ is not related to the thermodynamical notion of entropy. Firstly, it decreases over time, while the entropy increases in thermodynamics. However, there may be a link with the thermodynamical entropy for some particular systems. In the case of Eulerian gas dynamics, a natural mathematical entropy function is defined with respect to the thermodynamical entropy (cf. Section II.2 of (2)).

---

(1) R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1

(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996. doi:10.1007/978-1-4612-0713-9

share|cite|improve this answer

edited Aug 9 at 6:30

answered Aug 8 at 14:38

Harry49

4,6661825

add a comment | 

up vote
0
down vote

Let us consider the scalar one-dimensional case $u_t + f(u)_x = 0$ for sake of simplicity, as described in Section 3.8 of (1).

An entropy function $eta$ is a convex function of $u$ which satisfies the conservation law $eta(u)_t + psi(u)_x = 0$, where $psi$ is the entropy flux. In the definition, neither the flux $f$ nor the entropy flux $psi$ is convex, but only $eta$ is convex. This is a conventional choice with important consequences, "though one may consider a more general
framework" (2). If $eta$ is chosen convex, then the vanishing viscosity weak solution -- i.e., the physically relevant weak solution -- makes the entropy decrease in the weak sense. This leads to the definition of the entropy solution, which is a weak solution $u$ such that $eta(u)_t + psi(u)_x leq 0$ weakly for all convex entropy function $eta$.

As such, the definition of an entropy function $eta$ is not related to the thermodynamical notion of entropy. Firstly, it decreases over time, while the entropy increases in thermodynamics. However, there may be a link with the thermodynamical entropy for some particular systems. In the case of Eulerian gas dynamics, a natural mathematical entropy function is defined with respect to the thermodynamical entropy (cf. Section II.2 of (2)).

---

(1) R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1

(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996. doi:10.1007/978-1-4612-0713-9

share|cite|improve this answer

edited Aug 9 at 6:30

answered Aug 8 at 14:38

Harry49

4,6661825

add a comment | 

up vote
0
down vote

up vote
0
down vote

Let us consider the scalar one-dimensional case $u_t + f(u)_x = 0$ for sake of simplicity, as described in Section 3.8 of (1).

An entropy function $eta$ is a convex function of $u$ which satisfies the conservation law $eta(u)_t + psi(u)_x = 0$, where $psi$ is the entropy flux. In the definition, neither the flux $f$ nor the entropy flux $psi$ is convex, but only $eta$ is convex. This is a conventional choice with important consequences, "though one may consider a more general
framework" (2). If $eta$ is chosen convex, then the vanishing viscosity weak solution -- i.e., the physically relevant weak solution -- makes the entropy decrease in the weak sense. This leads to the definition of the entropy solution, which is a weak solution $u$ such that $eta(u)_t + psi(u)_x leq 0$ weakly for all convex entropy function $eta$.

As such, the definition of an entropy function $eta$ is not related to the thermodynamical notion of entropy. Firstly, it decreases over time, while the entropy increases in thermodynamics. However, there may be a link with the thermodynamical entropy for some particular systems. In the case of Eulerian gas dynamics, a natural mathematical entropy function is defined with respect to the thermodynamical entropy (cf. Section II.2 of (2)).

---

(1) R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1

(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996. doi:10.1007/978-1-4612-0713-9

share|cite|improve this answer

edited Aug 9 at 6:30

answered Aug 8 at 14:38

Harry49

4,6661825

Let us consider the scalar one-dimensional case $u_t + f(u)_x = 0$ for sake of simplicity, as described in Section 3.8 of (1).

An entropy function $eta$ is a convex function of $u$ which satisfies the conservation law $eta(u)_t + psi(u)_x = 0$, where $psi$ is the entropy flux. In the definition, neither the flux $f$ nor the entropy flux $psi$ is convex, but only $eta$ is convex. This is a conventional choice with important consequences, "though one may consider a more general
framework" (2). If $eta$ is chosen convex, then the vanishing viscosity weak solution -- i.e., the physically relevant weak solution -- makes the entropy decrease in the weak sense. This leads to the definition of the entropy solution, which is a weak solution $u$ such that $eta(u)_t + psi(u)_x leq 0$ weakly for all convex entropy function $eta$.

As such, the definition of an entropy function $eta$ is not related to the thermodynamical notion of entropy. Firstly, it decreases over time, while the entropy increases in thermodynamics. However, there may be a link with the thermodynamical entropy for some particular systems. In the case of Eulerian gas dynamics, a natural mathematical entropy function is defined with respect to the thermodynamical entropy (cf. Section II.2 of (2)).

---

(1) R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1

(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, 1996. doi:10.1007/978-1-4612-0713-9

share|cite|improve this answer

edited Aug 9 at 6:30

answered Aug 8 at 14:38

Harry49

4,6661825

share|cite|improve this answer

share|cite|improve this answer

edited Aug 9 at 6:30

edited Aug 9 at 6:30

edited Aug 9 at 6:30

answered Aug 8 at 14:38

Harry49

4,6661825

answered Aug 8 at 14:38

Harry49

4,6661825

answered Aug 8 at 14:38

Harry49

4,6661825

4,6661825

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