How to show that strictly hyperbolic implies strongly well posed? The 2019 Stack Overflow Developer Survey Results Are InEstimation on elliptic operatorShow that solution is in $C^infty(Omega)cap C(overlineOmega)$Description of a Space of FunctionsStability of semi-discrete approximations of initial-boundary value problemsShow that $d(z,z_1)$ is a metric.Find norm of $T:(ell^1,||cdot||_1)to(mathcal C[0,1],||cdot||_infty),$ $(T(xi))(x)=sum_k=0^infty a_kxi_k x^k,$ $xiinell^1$Resolvent InequalityProb. 10, Sec. 3.5, in Kreyszig's Functional Analysis: How to show that this set is at most countable?stein's complex analysis, functions of finite order.Fokker-Planck equation applied to $lvert xrvert^2$
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How to show that strictly hyperbolic implies strongly well posed?
The 2019 Stack Overflow Developer Survey Results Are InEstimation on elliptic operatorShow that solution is in $C^infty(Omega)cap C(overlineOmega)$Description of a Space of FunctionsStability of semi-discrete approximations of initial-boundary value problemsShow that $d(z,z_1)$ is a metric.Find norm of $T:(ell^1,||cdot||_1)to(mathcal C[0,1],||cdot||_infty),$ $(T(xi))(x)=sum_k=0^infty a_kxi_k x^k,$ $xiinell^1$Resolvent InequalityProb. 10, Sec. 3.5, in Kreyszig's Functional Analysis: How to show that this set is at most countable?stein's complex analysis, functions of finite order.Fokker-Planck equation applied to $lvert xrvert^2$
$begingroup$
Consider a first order system $partial_t u = P(D)u$ with $P$ given by
$$
P(xi) = sum_k=1^n iA_kxi_k.
$$
Here, $n$ denotes the dimension of the spatial domain of $u$ (i.e. $mathbbR^n$) and each $A_k in M_Ntimes N(mathbbC)$ for some $NinmathbbN$. More precisely, consider solutions $u : mathbbR^n times mathbbR to mathbbC^N$ to the system:
$$
partial_t u = P(D)u = sum_k=1^n A_k partial_ku.
$$
I want to show that if $P$ is strictly hyperbolic (i.e. diagonalizable with purely imaginary and distinct eigenvalues) then the system above is strongly well posed. Now, I've reduced this to showing that, there exist constants $C, a$ such that
$$
leftlvert e^tP(xi)right rvert leq Ce^a t
$$
for all $xiinmathbbR^n$ and $tgeq 0$. In the above, $leftlvert cdotrightrvert$ denotes the matrix norm. Does anyone have any advice on how to proceed?
functional-analysis analysis pde hyperbolic-equations
$endgroup$
This question has an open bounty worth +50
reputation from Community♦ ending ending at 2019-04-17 05:28:29Z">in 5 days.
This question has not received enough attention.
add a comment |
$begingroup$
Consider a first order system $partial_t u = P(D)u$ with $P$ given by
$$
P(xi) = sum_k=1^n iA_kxi_k.
$$
Here, $n$ denotes the dimension of the spatial domain of $u$ (i.e. $mathbbR^n$) and each $A_k in M_Ntimes N(mathbbC)$ for some $NinmathbbN$. More precisely, consider solutions $u : mathbbR^n times mathbbR to mathbbC^N$ to the system:
$$
partial_t u = P(D)u = sum_k=1^n A_k partial_ku.
$$
I want to show that if $P$ is strictly hyperbolic (i.e. diagonalizable with purely imaginary and distinct eigenvalues) then the system above is strongly well posed. Now, I've reduced this to showing that, there exist constants $C, a$ such that
$$
leftlvert e^tP(xi)right rvert leq Ce^a t
$$
for all $xiinmathbbR^n$ and $tgeq 0$. In the above, $leftlvert cdotrightrvert$ denotes the matrix norm. Does anyone have any advice on how to proceed?
functional-analysis analysis pde hyperbolic-equations
$endgroup$
This question has an open bounty worth +50
reputation from Community♦ ending ending at 2019-04-17 05:28:29Z">in 5 days.
This question has not received enough attention.
add a comment |
$begingroup$
Consider a first order system $partial_t u = P(D)u$ with $P$ given by
$$
P(xi) = sum_k=1^n iA_kxi_k.
$$
Here, $n$ denotes the dimension of the spatial domain of $u$ (i.e. $mathbbR^n$) and each $A_k in M_Ntimes N(mathbbC)$ for some $NinmathbbN$. More precisely, consider solutions $u : mathbbR^n times mathbbR to mathbbC^N$ to the system:
$$
partial_t u = P(D)u = sum_k=1^n A_k partial_ku.
$$
I want to show that if $P$ is strictly hyperbolic (i.e. diagonalizable with purely imaginary and distinct eigenvalues) then the system above is strongly well posed. Now, I've reduced this to showing that, there exist constants $C, a$ such that
$$
leftlvert e^tP(xi)right rvert leq Ce^a t
$$
for all $xiinmathbbR^n$ and $tgeq 0$. In the above, $leftlvert cdotrightrvert$ denotes the matrix norm. Does anyone have any advice on how to proceed?
functional-analysis analysis pde hyperbolic-equations
$endgroup$
Consider a first order system $partial_t u = P(D)u$ with $P$ given by
$$
P(xi) = sum_k=1^n iA_kxi_k.
$$
Here, $n$ denotes the dimension of the spatial domain of $u$ (i.e. $mathbbR^n$) and each $A_k in M_Ntimes N(mathbbC)$ for some $NinmathbbN$. More precisely, consider solutions $u : mathbbR^n times mathbbR to mathbbC^N$ to the system:
$$
partial_t u = P(D)u = sum_k=1^n A_k partial_ku.
$$
I want to show that if $P$ is strictly hyperbolic (i.e. diagonalizable with purely imaginary and distinct eigenvalues) then the system above is strongly well posed. Now, I've reduced this to showing that, there exist constants $C, a$ such that
$$
leftlvert e^tP(xi)right rvert leq Ce^a t
$$
for all $xiinmathbbR^n$ and $tgeq 0$. In the above, $leftlvert cdotrightrvert$ denotes the matrix norm. Does anyone have any advice on how to proceed?
functional-analysis analysis pde hyperbolic-equations
functional-analysis analysis pde hyperbolic-equations
asked Apr 8 at 1:40
user596383
This question has an open bounty worth +50
reputation from Community♦ ending ending at 2019-04-17 05:28:29Z">in 5 days.
This question has not received enough attention.
This question has an open bounty worth +50
reputation from Community♦ ending ending at 2019-04-17 05:28:29Z">in 5 days.
This question has not received enough attention.
add a comment |
add a comment |
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