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Let $B_R$ be a ball of radius $R>0$ centered at the origin in $\mathbb{R}^2$.

Consider the PDE $u_t + \nabla \cdot (u^2 \nabla \Delta u) = 0$, in $ (0,T)\times B_R,$ with boundary conditions $u =\frac{\partial}{\partial\nu} \Delta u = 0$, in $(0,T)\times B_R,$ and initial data $u(0,\cdot) = g,$ in $ B_R$, where for instance 

g[x] = 1

in $B_r$ and $0$ outside $B_r$, with $r < R$. A reference on numerical analysis for this problem is https://dml.cz/handle/10338.dmlcz/134511.

Is it possible to use Mathematica (for example NDSolve) to obtain and plot solutions to such equation?

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  • $\begingroup$ What have you tried? Have you looked into NDSolve and friends? $\endgroup$
    – MarcoB
    May 11, 2018 at 15:00
  • $\begingroup$ @MarcoB I don't know how to implement the boundary conditions on NDSolve or if using NDSolve is actually the correct approach. $\endgroup$
    – user58151
    May 11, 2018 at 15:11
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    $\begingroup$ That's a nonlinear PDE and Mathematica's FEM capacities don't support that at the moment (version 11.3). Moreover, it is a fourth order PDE and I doubt that these are supported at all.You could try reformulate this as a coupled system of order 2 and use NDSolve for solving linearized PDEs in order to implement a semi-implicit time integrator... $\endgroup$
    – Henrik Schumacher
    May 11, 2018 at 15:23
  • $\begingroup$ @HenrikSchumacher We can rewrite the PDE as $u_t + \nabla \cdot (u^2 \nabla w) = 0$ and $w = \Delta u$, but I don't know how to approach systems with Mathematica. $\endgroup$
    – user58151
    May 11, 2018 at 15:39
  • $\begingroup$ for fourth order PDE problems, We can use Isogeometric Analysis to solve it, but I am not sure, does Mathematica support Isogeometric Analysis? $\endgroup$
    – ABCDEMMM
    May 13, 2018 at 20:05

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