In the pictures below, the given velocity field in the adv-diff equation is smooth, the viscosity is a known constant, f is a given forcing term that can be neglected.
In the nonlinear Robin boundary condition, the parameters alpha, beta, lambda, and chi depend on time and space so that the boundary conditions on different segments of the domain boundary may be Dirichlet (alpha\ne 0, beta=0, lambda=0), Neumann (alpha=0, beta\ne 0, lambda=0), mixed linear Robin (alpha\ne 0, beta\ne 0, lambda=0), and nonlinear Robin (lambda\ne 0, chi\ne 0).
For the motivation and background info about this question, see the attached question.pdf.
The hard part is the boundary condition. The equation itself is linear but if $\nu$ depends on special variable or depends on $\phi$ then it is nonlinear. There are many article discuss this problem, google prorous media equation.
Janpou: thanks for the thoughts. The viscosity here is a constant. So the porous media equation does not apply here. The difficulty is only in the nonlinear Robin BC.
Qinghai, you can solve the differential equation exactly, hence, introduce unknown coefficients into the solution and apply the boundary condition. This will lead to a nonlinear equation, which may afford some type of closed-form solution, although, this is highly unlikely given the intractable nature of nonlinear equations and our inability to solve them. Regardless, you may linearize along some interval, find solutions close to some critical point or you might use numeric techniques to produce a series of approximations. Anyway, the problem for you is the BC and finding how this condition constrains the system.
Use w=u^2/2, then nonlinear BC may switch to nonlinear pde which would be easier.
Magnus: thanks for pointing out the book. The second edition just came out 2013. I will check it out.
Beware of Finite Difference methods: the numeric error for each step can often erase any significance in the output. Analytic closed-form methods are far superior.
It all depends what your goals really are (i.e., whether the system is even well-posed). Then there is also the issue about your BC appearing not to have being derived from first physical principles. The real question is how does one get the third term on the left-hand side of your boundary condition? This does not make much sense since your equation in the bulk is simply a reaction-advective equation. The point is for one to find conserved/dissipated quantities (say even in the case when u is divergent free). Just simply multiplying the equation by \phi and integrate by parts and there is already a problem. For instance, \beta should be a scalar multiple of \nu, and then assuming so, how do you get an energy estimate for the L^2 norm if the boundary term you get by integration is of higher--order and not of definite sign? Not even this quantity appears to be controlled. My opinion is that the physics does not seem justified here unless there is something that I am completely missing?
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