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I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation

[itex]\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2}[/itex] subject to the initial condition [itex]u(x,0)=g(x)[/itex] with addition boundary condition.

I saw in some references that the boundary condition is specified as [itex]u(0,t)=f(t)[/itex]. So there is only one boundary condition at the origin, while for completeness, I think there should be another boundary condition for the viscous term. What I suppose is that the implicit unsaid is [itex]u(\infty,t)=0[/itex]. But I don't know, it's a guess.

Could you who are experienced in PDE clarify what's the physical meaning of only specifying a boundary condition at the origin even for a high-order PDE? And to what extent is this formulation mathematically sound? Thank.

Jo