NYU Shanghai

We discuss $C^1$-regularity and developability of isometric immersions of flat domains into $\mathbb{R}^3$ enjoying a local fractional Sobolev $W^{1+s;2/s}$-regularity for $2/3 \leq s < 1$, generalising the known results on Sobolev (by Pakzad) and H\"{o}lder (by De Lellis--Pakzad) regimes. Ingredients of the proof include analysis of the weak Codazzi equations of isometric immersions, the study of $W^{1+s;2/s}$-gradient deformations with symmetric derivative and vanishing distributional Jacobian determinant, and the theory of compensated compactness. Joint work with M. Reza Pakzad and Armin Schikorra.

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position and its orientation vector, which lies on the unit sphere. We prove that, in the low temperature regime, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into the sphere. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals and the existence of its global weak solution was first obtained by Y.M Chen, using Ginzburg-Landau approximation.  The key ingredient of our result is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limiting local equilibrium distribution, as well as the energy dissipation estimates.  This is a joint work with Wei Wang in Zhejiang university.