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Simplified equation and added boundary condition. Typo.
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Atransportconfusion
Atransportconfusion
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I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$. One such example is: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f(x,y) \, , $$$$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = 1 \, , $$ with some appropriatethe (no inflow) boundary conditionscondition $u(\cdot,0) = u(0,\cdot) = 0$.

A simple analysis (method of characteristics etc.) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?

I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f(x,y) \, , $$ with some appropriate boundary conditions.

A simple analysis (method of characteristics etc.) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?

I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$. One such example is: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = 1 \, , $$ with the (no inflow) boundary condition $u(\cdot,0) = u(0,\cdot) = 0$.

A simple analysis (method of characteristics etc.) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?

Latex and a very very small touch of English
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edit approved Nov 6, 2017 at 21:26
Amir Sagiv
edit approved Nov 6, 2017 at 21:26
Amir Sagiv
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I have been studying the 2D time-independent advection equation on the unit square [0,1] x [0,1]$[0,1] \times [0,1]$: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f $$$$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f(x,y) \, , $$ with some appropriate boundary conditions.

A simple analysis (method of characteristics etc.) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?

I have been studying the 2D time-independent advection equation on the unit square [0,1] x [0,1]: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f $$ with some appropriate boundary conditions.

A simple analysis (method of characteristics etc) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?

I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f(x,y) \, , $$ with some appropriate boundary conditions.

A simple analysis (method of characteristics etc.) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?

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Atransportconfusion
Atransportconfusion
  • 11
  • 4

Advection equation regularity (2D and time independent)

I have been studying the 2D time-independent advection equation on the unit square [0,1] x [0,1]: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f $$ with some appropriate boundary conditions.

A simple analysis (method of characteristics etc) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?