\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A concept of solution and numerical experiments for forward-backward diffusion equations

Abstract / Introduction Related Papers Cited by
  • We study the gradient flow associated with the functional $F_\phi(u)$ := $\frac{1}{2}\int_{I} \phi(u_x)~dx$, where $\phi$ is non convex, and with its singular perturbation $F_\phi^\varepsilon(u)$:=$\frac{1}{2}\int_I (\varepsilon^2 (u_{x x})^2 + \phi(u_x))dx$. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions $u^\varepsilon$ of the singularly perturbed equation $u_t = - \varepsilon^2 u_{x x x x} + \frac{1}{2} \phi''(u_x)u_{x x}$ for small values of $\varepsilon>0$. Our analysis leads to a reinterpretation of the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$, and to a well defined notion of a solution. We also examine the conjecture that this solution coincides with the limit of $u^\varepsilon$ as $\varepsilon\to 0^+$.
    Mathematics Subject Classification: Primary: 35B25; Secondary: 47j06, 35K99.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(135) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return