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Integral form of the conservation law $u_t+f(u)_x=0$

0

1

$begingroup$

Consider the conservation law given by
$$u_t+f(u)_x=0$$
We know that in general weak solutions are not smooth but are bounded in $L^infty$ norm (they do not belong to Sobolev spaces).

However while deriving the numerical schemes most of the books say, integrating the conservation law over $(a,b) times (t_1,t_2)$ and applying fundamental theorem of calculus we get
$$int_a^b u(x,t_1)dx - int_a^b u(x,t_2)dx= -int_t_1^t_2 f(u(b,t)dt+ int_t_1^t_2 f(u(a,t)dt$$
I have the following doubts:

1. How can we perform integration by parts as the solution does not possess any regularity?




2. If a function satisfies the above integral formulation, can we say that it is a weak solution? Conversely, if $u$ is a weak solution, will it satisfy the above integral formulation?

Thank you.

analysis pde numerical-methods hyperbolic-equations

share|cite|improve this question

edited Apr 8 at 13:43

Andrews

1,2962423

asked Apr 4 at 12:39

RosyRosy

1476

$endgroup$

add a comment | 

0

1

$begingroup$

Consider the conservation law given by
$$u_t+f(u)_x=0$$
We know that in general weak solutions are not smooth but are bounded in $L^infty$ norm (they do not belong to Sobolev spaces).

However while deriving the numerical schemes most of the books say, integrating the conservation law over $(a,b) times (t_1,t_2)$ and applying fundamental theorem of calculus we get
$$int_a^b u(x,t_1)dx - int_a^b u(x,t_2)dx= -int_t_1^t_2 f(u(b,t)dt+ int_t_1^t_2 f(u(a,t)dt$$
I have the following doubts:

1. How can we perform integration by parts as the solution does not possess any regularity?




2. If a function satisfies the above integral formulation, can we say that it is a weak solution? Conversely, if $u$ is a weak solution, will it satisfy the above integral formulation?

Thank you.

analysis pde numerical-methods hyperbolic-equations

share|cite|improve this question

edited Apr 8 at 13:43

Andrews

1,2962423

asked Apr 4 at 12:39

RosyRosy

1476

$endgroup$

add a comment | 

$begingroup$

Consider the conservation law given by
$$u_t+f(u)_x=0$$
We know that in general weak solutions are not smooth but are bounded in $L^infty$ norm (they do not belong to Sobolev spaces).

However while deriving the numerical schemes most of the books say, integrating the conservation law over $(a,b) times (t_1,t_2)$ and applying fundamental theorem of calculus we get
$$int_a^b u(x,t_1)dx - int_a^b u(x,t_2)dx= -int_t_1^t_2 f(u(b,t)dt+ int_t_1^t_2 f(u(a,t)dt$$
I have the following doubts:

1. How can we perform integration by parts as the solution does not possess any regularity?




2. If a function satisfies the above integral formulation, can we say that it is a weak solution? Conversely, if $u$ is a weak solution, will it satisfy the above integral formulation?

Thank you.

analysis pde numerical-methods hyperbolic-equations

share|cite|improve this question

edited Apr 8 at 13:43

Andrews

1,2962423

asked Apr 4 at 12:39

RosyRosy

1476

$endgroup$

share|cite|improve this question

edited Apr 8 at 13:43

Andrews

1,2962423

asked Apr 4 at 12:39

RosyRosy

1476

share|cite|improve this question

edited Apr 8 at 13:43

Andrews

1,2962423

asked Apr 4 at 12:39

RosyRosy

1476

edited Apr 8 at 13:43

Andrews

1,2962423

edited Apr 8 at 13:43

Andrews

1,2962423

asked Apr 4 at 12:39

RosyRosy

1476

asked Apr 4 at 12:39

RosyRosy

1476