Skip to main content
MathOverflow

Return to Question

clarification
Source Link
Dorian
Dorian
  • 2.6k
  • 23
  • 32

I've prove existence using the Galerkin method forBurger's equation with dissipation: $u_t + uu_x - u_xx = 0$$u_t + uu_x - u_{xx} = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

Clarification: I have proven existence for $u \in L^2([0,T];H_0^1(\Omega))$, $u_t \in L^2([0,T];H^{-1}(\Omega))$.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed concerning $L^2$ regularity for inviscid Burger's equation? Or perhaps I'm missing something obvious?

I've prove existence using the Galerkin method forBurger's equation with dissipation: $u_t + uu_x - u_xx = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed concerning $L^2$ regularity for inviscid Burger's equation? Or perhaps I'm missing something obvious?

I've prove existence using the Galerkin method forBurger's equation with dissipation: $u_t + uu_x - u_{xx} = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

Clarification: I have proven existence for $u \in L^2([0,T];H_0^1(\Omega))$, $u_t \in L^2([0,T];H^{-1}(\Omega))$.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed concerning $L^2$ regularity for inviscid Burger's equation? Or perhaps I'm missing something obvious?

copy edit
Source Link
Charles Matthews
Charles Matthews
  • 12.6k
  • 35
  • 64

A priori energy estimates for burger'sBurger's equation with dissipation.

I've prove existence using the Galerkin method to the burger'sforBurger's equation with dissipation: $u_t + uu_x - u_xx = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed conerningconcerning $L^2$ regularity for inviscid burger'sBurger's equation? Or perhaps I'm missing something obvious?

A priori energy estimates for burger's equation with dissipation.

I've prove existence using the Galerkin method to the burger's equation with dissipation: $u_t + uu_x - u_xx = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed conerning $L^2$ regularity for inviscid burger's equation? Or perhaps I'm missing something obvious?

A priori energy estimates for Burger's equation with dissipation

I've prove existence using the Galerkin method forBurger's equation with dissipation: $u_t + uu_x - u_xx = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed concerning $L^2$ regularity for inviscid Burger's equation? Or perhaps I'm missing something obvious?

Source Link
Dorian
Dorian
  • 2.6k
  • 23
  • 32

A priori energy estimates for burger's equation with dissipation.

I've prove existence using the Galerkin method to the burger's equation with dissipation: $u_t + uu_x - u_xx = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed conerning $L^2$ regularity for inviscid burger's equation? Or perhaps I'm missing something obvious?