Weak formulation for non-linear Laplace equation with prefactor

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I have two coupled equations which I would like to rewrite into the weak form:
$$beginalign
partial_tA&=ldots\
partial_tB&=Anablaleft(c(B)nabla Bright)
endalign$$
Equation $A$ is not of importance, thus is not written fully. Usually, for an equation of the form
$$partial_tC=nablaleft(knabla Cright)$$
I can write the weak form (by neglecting the boundaries) with $phi$ as test function as
$$partial_tC=-int knabla Cnablaphi dx$$
Now I can not do that partial integration due to the prefactor $A$ in the equation above. Do I thus have to write the weak form as
$$partial_tB=int Anabla(c(B)nabla B))phi dx$$
or do I have to do partial integration (which gets messy quite fast), or is there another way I could use to simplify the weak form?




integration pde




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    I have two coupled equations which I would like to rewrite into the weak form:
    $$beginalign
    partial_tA&=ldots\
    partial_tB&=Anablaleft(c(B)nabla Bright)
    endalign$$
    Equation $A$ is not of importance, thus is not written fully. Usually, for an equation of the form
    $$partial_tC=nablaleft(knabla Cright)$$
    I can write the weak form (by neglecting the boundaries) with $phi$ as test function as
    $$partial_tC=-int knabla Cnablaphi dx$$
    Now I can not do that partial integration due to the prefactor $A$ in the equation above. Do I thus have to write the weak form as
    $$partial_tB=int Anabla(c(B)nabla B))phi dx$$
    or do I have to do partial integration (which gets messy quite fast), or is there another way I could use to simplify the weak form?




    integration pde




    share|cite|improve this question























      up vote
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      up vote
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      down vote

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      I have two coupled equations which I would like to rewrite into the weak form:
      $$beginalign
      partial_tA&=ldots\
      partial_tB&=Anablaleft(c(B)nabla Bright)
      endalign$$
      Equation $A$ is not of importance, thus is not written fully. Usually, for an equation of the form
      $$partial_tC=nablaleft(knabla Cright)$$
      I can write the weak form (by neglecting the boundaries) with $phi$ as test function as
      $$partial_tC=-int knabla Cnablaphi dx$$
      Now I can not do that partial integration due to the prefactor $A$ in the equation above. Do I thus have to write the weak form as
      $$partial_tB=int Anabla(c(B)nabla B))phi dx$$
      or do I have to do partial integration (which gets messy quite fast), or is there another way I could use to simplify the weak form?




      integration pde




      share|cite|improve this question













      I have two coupled equations which I would like to rewrite into the weak form:
      $$beginalign
      partial_tA&=ldots\
      partial_tB&=Anablaleft(c(B)nabla Bright)
      endalign$$
      Equation $A$ is not of importance, thus is not written fully. Usually, for an equation of the form
      $$partial_tC=nablaleft(knabla Cright)$$
      I can write the weak form (by neglecting the boundaries) with $phi$ as test function as
      $$partial_tC=-int knabla Cnablaphi dx$$
      Now I can not do that partial integration due to the prefactor $A$ in the equation above. Do I thus have to write the weak form as
      $$partial_tB=int Anabla(c(B)nabla B))phi dx$$
      or do I have to do partial integration (which gets messy quite fast), or is there another way I could use to simplify the weak form?





      integration pde





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      edited Jul 29 at 16:32









      Bernard

      110k635102




      110k635102









      asked Jul 29 at 15:57









      arc_lupus

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