Compensated Compactness And Conservation laws












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I am trying to understand Compensated compactness. I am new to this area. I have the following doubts to start with



1) I have been reading many books where its been written in differnet ways. So What is compensated compactness? is it a name of a particular theorem or its a general term used for a kind of compactness result?



2) How does it help in understanding conservation laws.?



3) Is there any book/notes which briefly explains compensated compactness which is good enough to understand conservation laws.



4)I read that



"If $(u^n_1, u^n_2,....u^n_k) rightarrow(u_1, u_2,....u_k)$ and
$(v^n_1, v^n_2,....v^n_k) rightarrow(v_1, v_2,....v_k)$ weakly in $L^2$. Such that $operatorname{div}textbf{u^n}$ and $operatorname{curl} textbf{v^n}$ are bounded in $L^2$ then $sum_{i=1}^k u^n_iv^n_i rightarrow sum_{i=1}^k u_iv_i$ in the sense of distribution"



Is this result called compesated compactness? If so why the name compensated compactness? what did we compensate here?



Thanks










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    0














    I am trying to understand Compensated compactness. I am new to this area. I have the following doubts to start with



    1) I have been reading many books where its been written in differnet ways. So What is compensated compactness? is it a name of a particular theorem or its a general term used for a kind of compactness result?



    2) How does it help in understanding conservation laws.?



    3) Is there any book/notes which briefly explains compensated compactness which is good enough to understand conservation laws.



    4)I read that



    "If $(u^n_1, u^n_2,....u^n_k) rightarrow(u_1, u_2,....u_k)$ and
    $(v^n_1, v^n_2,....v^n_k) rightarrow(v_1, v_2,....v_k)$ weakly in $L^2$. Such that $operatorname{div}textbf{u^n}$ and $operatorname{curl} textbf{v^n}$ are bounded in $L^2$ then $sum_{i=1}^k u^n_iv^n_i rightarrow sum_{i=1}^k u_iv_i$ in the sense of distribution"



    Is this result called compesated compactness? If so why the name compensated compactness? what did we compensate here?



    Thanks










    share|cite|improve this question

























      0












      0








      0


      1





      I am trying to understand Compensated compactness. I am new to this area. I have the following doubts to start with



      1) I have been reading many books where its been written in differnet ways. So What is compensated compactness? is it a name of a particular theorem or its a general term used for a kind of compactness result?



      2) How does it help in understanding conservation laws.?



      3) Is there any book/notes which briefly explains compensated compactness which is good enough to understand conservation laws.



      4)I read that



      "If $(u^n_1, u^n_2,....u^n_k) rightarrow(u_1, u_2,....u_k)$ and
      $(v^n_1, v^n_2,....v^n_k) rightarrow(v_1, v_2,....v_k)$ weakly in $L^2$. Such that $operatorname{div}textbf{u^n}$ and $operatorname{curl} textbf{v^n}$ are bounded in $L^2$ then $sum_{i=1}^k u^n_iv^n_i rightarrow sum_{i=1}^k u_iv_i$ in the sense of distribution"



      Is this result called compesated compactness? If so why the name compensated compactness? what did we compensate here?



      Thanks










      share|cite|improve this question













      I am trying to understand Compensated compactness. I am new to this area. I have the following doubts to start with



      1) I have been reading many books where its been written in differnet ways. So What is compensated compactness? is it a name of a particular theorem or its a general term used for a kind of compactness result?



      2) How does it help in understanding conservation laws.?



      3) Is there any book/notes which briefly explains compensated compactness which is good enough to understand conservation laws.



      4)I read that



      "If $(u^n_1, u^n_2,....u^n_k) rightarrow(u_1, u_2,....u_k)$ and
      $(v^n_1, v^n_2,....v^n_k) rightarrow(v_1, v_2,....v_k)$ weakly in $L^2$. Such that $operatorname{div}textbf{u^n}$ and $operatorname{curl} textbf{v^n}$ are bounded in $L^2$ then $sum_{i=1}^k u^n_iv^n_i rightarrow sum_{i=1}^k u_iv_i$ in the sense of distribution"



      Is this result called compesated compactness? If so why the name compensated compactness? what did we compensate here?



      Thanks







      functional-analysis pde weak-convergence hyperbolic-equations






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      share|cite|improve this question











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      asked Dec 26 '18 at 18:30









      Rosy

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