# Research

*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:

- (Very painstakingly!) Correcting an error in the coefficients of the
critical eigenvalue of the
linearized spectral problem of the 2D Swift Hohenberg equation, as
originally found in (Mielke
96) and
extending said coefficients to a higher level of accuracy.
- Working out and proving the $L^1-L^\infty$ bounds of the nonlinear terms
of our Split Swift-Hohenberg equation.
- Putting together the final details of the contraction argument to prove
nonlinear diffusivity.

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.