In this paper we establish the local Lyapunov property of certain $\mathrm{L}^{p}$ and Besov norms of the vorticity fields. We have resolved in part, a certain open problem posed by Tosio Kato for the three-dimensional Navier-Stokes equation by studying the vorticity equation. The local dissipativity of the sum of linear and non-linear operators of the vorticity equation is established. One of the main techniques used here is Littlewood-Paley analysis.

Travelling fronts for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity $\varepsilon \geq 0$ exist for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both in the parabolic case when $\varepsilon >0$ and in the hyperbolic case when $\varepsilon =0$. We prove that as $\varepsilon \rightarrow 0$, the minimal velocity ${c_{\varepsilon}^*}$ converges to $c^*$, the minimal value when $\varepsilon =0$, and that ${c_{\varepsilon}^*}\geq c^*$ for all $\varepsilon >0$. The proof uses comparison theorems and the variational characterization of the minimal parabolic front velocity. This convergence also yields a reaction-independent sufficient condition for the minimal velocity of the parabolic problem for small positive $\varepsilon$ to be strictly greater than the value predicted by the problem linearized about the unstable equilibrium, that is, for the minimal-velocity travelling front of the viscous equation to be pushed for sufficiently small $\varepsilon$.

We consider degenerate viscous shock profiles arising in systems of two regularized conservation laws, where degeneracy here describes viscous shock profiles for which the asymptotic endstates are sonic to the associated hyperbolic system (the shock speed is equal to one of the characteristic speeds). Proceeding with pointwise estimates on the Green's function for the linear system of equations that arises upon linearization of the conservation law about a degenerate viscous shock profile, we establish that spectral stability, defined in terms of an Evans function, implies nonlinear stability. The asymptotic rate of decay for the perturbation is found both pointwise and in all $L^p$ norms, $p \ge 1$.

Using Struwe's "monotonicity trick" and the recent blow-up analysis of Ohtsuka and Suzuki, we prove the existence of mountain-pass solutions to a mean field equation arising in two-dimensional turbulence.

We establish oscillation criteria for solutions of quasilinear second order elliptic equations. We do not impose any additional conditions on the nonlinear terms except for the continuity. In particular, we can characterize the oscillation property of every solution for autonomous equations.