Question

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation \begin{equation} -\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0. \end{equation} Here the force $f \in L^2$, and $div u = \sum_j \partial u/\partial x_j $. Is there any solutions of the equation in $H^1$? And how to prove it. Thanks for any hints and references.

\textbf{Edit}: In fact, the $1-d$ equation we considered is the stationary equation of the BBM equation $$ u_t - u_{txx} - u_{xx} + u^2u_x = f. $$

Answer 1

Firstly, turning to the $\mathbb{R}^1$ case, we note that $$ -u_{xx}+u^2u_x=-\partial_x\left(u_x-\frac{1}{3}u^3\right)=f(x) $$ and so we get a Chini equation (see http://mathoverflow.net/questions/104006/inhomogeneous-bernoulli-equation): $$ -u_x+\frac{1}{3}u^3=\int_{x_0}^xf(x')dx'+C $$ and a general solution to this equation is not known. Turning to $\mathbb{R}^3$, you can see that your equation can be written down in the form $$ \nabla\cdot {\bf v}=-f $$ being $$ {\bf v}=\left(u_x-\frac{1}{3}u^3,u_y-\frac{1}{3}u^3,u_z-\frac{1}{3}u^3\right) $$ and you have to satisfy the Gauss identity $$ \int_{\partial\Omega}{\bf v}\cdot d{\bf S}=-\int_{\Omega}dV f. $$ This means that one has to find three functions $v_1,\ v_2,\ v_3$ that satisfy three different Chini equations. Again, a general solution misses in this case. The conclusion to draw is that you need to specialize the problem to see if this becomes manageable in some case or resort to some perturbation technique.