Note: I’ve been told that the math doesn’t render nicely on mobile. This is a general failing of MathJaX not being responsive, which isn’t ideal, but currently there isn’t an easy way to avoid it.

Diffusive Stability of the Swift-Hohenberg equation near the Zigzag Boundary.

During a Math REU @ Ohio University with Qiliang Wu, I, along with Mason Haberle and Professor Wu, researched the diffusive stability of the 2D Swift-Hohenberg equation, that is:

Theorem: Given $0 < \varepsilon^2 \ll 1$, the roll solution $u_p(k_z x_1; k_z)$, where the zigzag wave number $k_z(\varepsilon) = 1 - \frac{\varepsilon^4}{512} + h.o.t.$, of the 2D SHE

\[u_t = \left[-(1 + \Delta_x)^2 + \varepsilon^2\right]u - u^3\]

is nonlinearly stable. That is, given initial pertubation from $u_p$ in the form $u_0 = u_p + v_0$ where:

\[||\hat{v}_0||_{L^1} + ||\hat{v}_0||_{L^\infty} \ll 1\]

then we find that $\mid\mid v(\cdot,t)\mid\mid_{L^{\infty}} = O ( t^{-3/4} )$.

My main contributions were in:

The paper is still in draft stages, and thus I cannot upload a sufficiently polished copy here, but the topic was presented by by Qiliang at Miami Universities’ Forty-Seventh Annual Conference in Differential Equations and Dynamical Systems and their Applications. Here are the slides.