Compensated Compactness And Conservation laws
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
asked Dec 26 '18 at 18:30
Rosy
1044
add a comment |
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
asked Dec 26 '18 at 18:30
Rosy
1044
add a comment |
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
asked Dec 26 '18 at 18:30
Rosy
1044
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
functional-analysis pde weak-convergence hyperbolic-equations
asked Dec 26 '18 at 18:30
Rosy
1044
asked Dec 26 '18 at 18:30
Rosy
1044
asked Dec 26 '18 at 18:30
Rosy
1044
asked Dec 26 '18 at 18:30
Rosy
1044
asked Dec 26 '18 at 18:30
Rosy
1044
1044
add a comment |
add a comment |
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