Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials

作者：Eden A^*; Kalantarov V K; Zelik S V

来源：Journal of Mathematical Physics, 2013, 54(4): 041502.

DOI：10.1063/1.4798786

We study initial boundary value problems for the unstable convective Cahn-Hilliard (CH) equation, i.e., the Cahn Hilliard equation whose energy integral is not bounded below. It is well-known that without the convective term, the solutions of the unstable CH equation. partial derivative(t)u + partial derivative(4)(x)u + partial derivative(2)(x)(vertical bar u broken vertical bar(p)u) = 0 may blow up in finite time for any p %26gt; 0. In contrast to that, we show that the presence of the convective term u partial derivative(x)u in the Cahn-Hilliard equation prevents blow up at least for 0 %26lt; p %26lt; 4/9. We also show that the blowing up solutions still exist if p is large enough (p %26gt;= 2). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.

• 出版日期2013-4