In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the Lp-norm for all p ≧ 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in Lp (P ≧ 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.

Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the 3D finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in 3D, which do exist for the same system in 2D.

In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels $\Pi_{k, [E_1, E_2]} (z,w)$ and the associated DOS (density of states) $\Pi_{k, [E_1, E_2]} (z)$ across the interface $\mathcal{C}$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H : M \to \mathbb{R}$ on a Kähler manifold. The allowed region is $H^{-1} ([E_1, E_2])$ and the interface $\mathcal{C}$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-\frac{1}{2}}$ tube around $\mathcal{C}$ for regular interfaces, one obtains $\delta$-asymptotics in $k^{-\frac{1}{4}}$ tubes around singular points of a critical interface. In $k^{-\frac{1}{2}}$ tubes, the transition law is given by the osculating metaplectic propagator.

The dynamic stability of vortex solutions to the Ginzburg-Landau and nonlinear Schrdinger equations is the basic assumption of the asymptotic particle plus field description of interacting vortices. For the Ginzburg-Landau dynamics we prove that all vortices are asymptotically nonlinearly stable relative to small radial perturbations. Initially finite energy perturbations of vortices decay to zero in<i>L</i> ^<i>p</i>(²) spaces with an algebraic rate as time tends to infinity. We also prove that under general (nonradial) perturbations, the plus and minus one-vortices are linearly dynamically stable in<i>L</i> ²; the linearized operator has spectrum equal to (, 0] and generates a<i>C</i> ₀ semigroup of contractions on<i>L</i> ²(²). The nature of the zero energy point is clarified; it is<i>resonance</i>, a property related to the infinite energy of planar vortices. Our results on the linearized operator are also used to show that the plus and

We study the translational and rotational dynamics of neutrally buoyant finite-size spheroids in hydrodynamic turbulence by means of fully resolved numerical simulations. We examine axisymmetric shapes, from oblate to prolate, and the particle volume dependences. We show that the accelerations and rotations experienced by non-spherical inertial-scale particles result from volume filtered fluid forces and torques, similar to spherical particles. However, the particle orientations carry signatures of preferential alignments with the surrounding flow structures, which are reflected in distinct axial and lateral fluctuations for accelerations and rotation rates. The randomization of orientations does not occur even for particles with volume-equivalent diameter size in the inertial range, here up to 60 dissipative units (eta) at Taylor-scale Reynolds number Re-lambda = 120. Additionally, we demonstrate that the role of fluid boundary layers around the particles cannot be neglected in reaching a quantitative understanding of particle statistical dynamics, as they affect the intensities of the angular velocities and the relative importance of tumbling with respect to spinning rotations. This study brings to the fore the importance of inertial-scale flow structures in homogeneous and isotropic turbulence and their impacts on the transport of neutrally buoyant bodies with sizes in the inertial range.