Solvability of a nonlinear elliptic equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:57:20Z http://mathoverflow.net/feeds/question/109289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109289/solvability-of-a-nonlinear-elliptic-equation Solvability of a nonlinear elliptic equation Wang Ming 2012-10-10T11:13:38Z 2012-10-12T12:17:21Z <p>Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation $$-\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0.$$ 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.</p> <p>\textbf{Edit}: In fact, the $1-d$ equation we considered is the stationary equation of the BBM equation $$u_t - u_{txx} - u_{xx} + u^2u_x = f.$$ </p> http://mathoverflow.net/questions/109289/solvability-of-a-nonlinear-elliptic-equation/109455#109455 Answer by Jon for Solvability of a nonlinear elliptic equation Jon 2012-10-12T12:17:21Z 2012-10-12T12:17:21Z <p>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 <a href="http://mathoverflow.net/questions/104006/inhomogeneous-bernoulli-equation" rel="nofollow">http://mathoverflow.net/questions/104006/inhomogeneous-bernoulli-equation</a>): $$-u_x+\frac{1}{3}u^3=\int_{x_0}^xf(x')dx'+C$$ 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.</p>