Search

We establish a sharp regularity result for the normal trace of the solution to the anisotropic linear elasticity system with Dirichlet boundary condition on a Lipschitz domain. Using this result we obtain a new existence result for a fluid-structure interaction model in the case when the structure is an anisotropic elastic body.

We address the problem of analyticity of smooth solutions u of the incompressible Euler equations. If the initial datum is real-analytic, the solution remains real-analytic as long as f t 0 ∥∇u(·, s)∥ L∞ ds < ∞. Using a Gevrey-class approach we obtain lower bounds on the radius of space analyticity which depend algebraically on exp ∫ t 0 ∥∇u(·,s)∥ L∞ ds. In particular, we answer in the positive a question posed by Levermore and Oliver.

We study backward uniqueness properties for equations of the form u' + Au = f. Under mild regularity assumptions on A and f, it is shown that u(0) = 0 implies u(t) = 0 for t < 0. The argument is based on α-log and log-log convexity. The results apply to mildly nonlinear parabolic equations and systems with rough coefficients and the 2D Navier-Stokes system.

We address the backward uniqueness property for the equation u t - Δu = w j ∂ j u + vu in R x (T 0 ,0]. We show that under rather general conditions on v and w, u| t=0 = 0 implies that u vanishes to infinite order for all points (x,0). It follows that the backward uniqueness holds if w = 0 and v ∈ L∞([0,T 0 ], L p (R n )) when p > n/2. The borderline case p = n/2 is also covered with an additional continuity and smallness assumption.

We show that the normal derivative of a harmonic function which vanishes on an open subset of the boundary of a Dini domain cannot vanish on a subset of positive surface measure.