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Heat equation - stationary solution


The initial condition for a heat equation with stationary solution subtractedIs the parabolic heat equation with pure neumann conditions well posed?Monotone solutions of the heat equation1D Rod, Heat EquationFinding steady-state solution for two-dimensional heat equationDifferential equation with homogeneous Dirichlet-Neumann boundary conditions2D Heat Equation - Exact Solution$L^infty$ norm of the heat equation solution, with Neumann conditionsHeat equation with time-dependent transport termIs the steady state solution of the Heat Equation with Dirichlet boundary conditions always 0?













0












$begingroup$


What are the conditions for the solution of the heat equation to become [sufficiently] stationary, i.e. to not change any more over time?



Here, I am talking about the 3D heat equation with Neumann and Dirichlet boundary data:
$$ partial_t u - a Delta u = f $$
with initial condition
$$ u(x,0) = u_0(x) $$
and boundary conditions
$$ u = g $$ and
$$ partial_n u = q $$
on some part of the boundary each.



Let's assume that $f,u_0,g,q$ and the unspecified domain are "sufficiently nice" for our problem. I am more looking for a broad answer regarding the convergence to the steady-state solution, as I observed this behaviour during some experiments.



Some thoughts of mine so far:



I guess that $f,g,q$ cannot change over time, to allow a steady-state solution? Or do I even need some form of $q=0$?



Or is this a simple matter of solving the new elliptic problem resulting from $partial_t u = 0$



A followup question would be: Can one do a time estimate for when such a solution might be reached?










share|cite|improve this question









$endgroup$











  • $begingroup$
    You are correct that $f,g, q$ must all become constant with respect to time, since all three can be expressed in terms of $u$, if $u$ is not time-dependent, they cannot be either. I suspect that is also a sufficient condition. If there is nothing exciting the system, I would expect it to enter a steady-state. But I don't have a mathematical demonstration.
    $endgroup$
    – Paul Sinclair
    Mar 16 at 20:21















0












$begingroup$


What are the conditions for the solution of the heat equation to become [sufficiently] stationary, i.e. to not change any more over time?



Here, I am talking about the 3D heat equation with Neumann and Dirichlet boundary data:
$$ partial_t u - a Delta u = f $$
with initial condition
$$ u(x,0) = u_0(x) $$
and boundary conditions
$$ u = g $$ and
$$ partial_n u = q $$
on some part of the boundary each.



Let's assume that $f,u_0,g,q$ and the unspecified domain are "sufficiently nice" for our problem. I am more looking for a broad answer regarding the convergence to the steady-state solution, as I observed this behaviour during some experiments.



Some thoughts of mine so far:



I guess that $f,g,q$ cannot change over time, to allow a steady-state solution? Or do I even need some form of $q=0$?



Or is this a simple matter of solving the new elliptic problem resulting from $partial_t u = 0$



A followup question would be: Can one do a time estimate for when such a solution might be reached?










share|cite|improve this question









$endgroup$











  • $begingroup$
    You are correct that $f,g, q$ must all become constant with respect to time, since all three can be expressed in terms of $u$, if $u$ is not time-dependent, they cannot be either. I suspect that is also a sufficient condition. If there is nothing exciting the system, I would expect it to enter a steady-state. But I don't have a mathematical demonstration.
    $endgroup$
    – Paul Sinclair
    Mar 16 at 20:21













0












0








0





$begingroup$


What are the conditions for the solution of the heat equation to become [sufficiently] stationary, i.e. to not change any more over time?



Here, I am talking about the 3D heat equation with Neumann and Dirichlet boundary data:
$$ partial_t u - a Delta u = f $$
with initial condition
$$ u(x,0) = u_0(x) $$
and boundary conditions
$$ u = g $$ and
$$ partial_n u = q $$
on some part of the boundary each.



Let's assume that $f,u_0,g,q$ and the unspecified domain are "sufficiently nice" for our problem. I am more looking for a broad answer regarding the convergence to the steady-state solution, as I observed this behaviour during some experiments.



Some thoughts of mine so far:



I guess that $f,g,q$ cannot change over time, to allow a steady-state solution? Or do I even need some form of $q=0$?



Or is this a simple matter of solving the new elliptic problem resulting from $partial_t u = 0$



A followup question would be: Can one do a time estimate for when such a solution might be reached?










share|cite|improve this question









$endgroup$




What are the conditions for the solution of the heat equation to become [sufficiently] stationary, i.e. to not change any more over time?



Here, I am talking about the 3D heat equation with Neumann and Dirichlet boundary data:
$$ partial_t u - a Delta u = f $$
with initial condition
$$ u(x,0) = u_0(x) $$
and boundary conditions
$$ u = g $$ and
$$ partial_n u = q $$
on some part of the boundary each.



Let's assume that $f,u_0,g,q$ and the unspecified domain are "sufficiently nice" for our problem. I am more looking for a broad answer regarding the convergence to the steady-state solution, as I observed this behaviour during some experiments.



Some thoughts of mine so far:



I guess that $f,g,q$ cannot change over time, to allow a steady-state solution? Or do I even need some form of $q=0$?



Or is this a simple matter of solving the new elliptic problem resulting from $partial_t u = 0$



A followup question would be: Can one do a time estimate for when such a solution might be reached?







pde heat-equation






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Mar 16 at 10:19









mh333mh333

1




1











  • $begingroup$
    You are correct that $f,g, q$ must all become constant with respect to time, since all three can be expressed in terms of $u$, if $u$ is not time-dependent, they cannot be either. I suspect that is also a sufficient condition. If there is nothing exciting the system, I would expect it to enter a steady-state. But I don't have a mathematical demonstration.
    $endgroup$
    – Paul Sinclair
    Mar 16 at 20:21
















  • $begingroup$
    You are correct that $f,g, q$ must all become constant with respect to time, since all three can be expressed in terms of $u$, if $u$ is not time-dependent, they cannot be either. I suspect that is also a sufficient condition. If there is nothing exciting the system, I would expect it to enter a steady-state. But I don't have a mathematical demonstration.
    $endgroup$
    – Paul Sinclair
    Mar 16 at 20:21















$begingroup$
You are correct that $f,g, q$ must all become constant with respect to time, since all three can be expressed in terms of $u$, if $u$ is not time-dependent, they cannot be either. I suspect that is also a sufficient condition. If there is nothing exciting the system, I would expect it to enter a steady-state. But I don't have a mathematical demonstration.
$endgroup$
– Paul Sinclair
Mar 16 at 20:21




$begingroup$
You are correct that $f,g, q$ must all become constant with respect to time, since all three can be expressed in terms of $u$, if $u$ is not time-dependent, they cannot be either. I suspect that is also a sufficient condition. If there is nothing exciting the system, I would expect it to enter a steady-state. But I don't have a mathematical demonstration.
$endgroup$
– Paul Sinclair
Mar 16 at 20:21










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