In this paper, we investigate the integrability and asymptotic behaviors of positive solutions for a nonlinear integral system in a functional setting. Using the regularity lifting lemma and some delicate analysis techniques, we obtain the optimal integral intervals and the asymptotic estimates for such solutions around the origin and near infinity. Moreover, the index of regular solution is distinct from one of the previous several related systems.

We present results on breather solutions of a discrete nonlinear Schrodinger equation with a cubic Hartree-type nonlinearity that models laser light propagation in waveguide arrays that use a nematic liquid crystal substratum. A recent study of that model by Ben et al showed that nonlocality leads to some novel properties such as the existence of orbitaly stable breathers with internal modes, and of shelf-like configurations with maxima at the interface. In this work we present rigorous results on these phenomena and consider some more general solutions. First, we study energy minimizing breathers, showing existence as well as symmetry and monotonicity properties. We also prove results on the spectrum of the linearization around one-peak breathers, solutions that are expected to coincide with minimizers in the regime of small linear intersite coupling. A second set of results concerns shelf-type breather solutions that may be thought of as limits of solutions examined in Ben et. al. We show the existence of solutions with a non-monotonic front-like shape and justify computations of the essential spectrum of the linearization around these solutions in the local and nonlocal cases.

We prove the existence of weak solutions to a kinetic flocking model with cut-off interaction function by using the weak convergence method. Under the natural assumption that the v-support of the initial distribution function f_0(x,v) is bounded, we show that the v-support of the distribution function f(t,x,v) is uniformly bounded in time. Employing this property, we remove the constraint in the paper of Karper, Mellet and Trivisa[SIAM. J. Math. Anal., (45)2013, pp.215-243] that the initial distribution function should have better integrability for large |x|.

In this paper, we study the uniform regularity and vanishing viscosity limit for the compressible nematic liquid crystal flows in three dimensional bounded domain. One establishes the uniform estimates for the solutions in a conormal Sobolev space and obtains the uniform estimates for the density and velocity in W^{1,\infty}. Then, it is shown that there exists a unique strong solution for the compressible nematic liquid crystal flows in a finite time interval which is independent of the viscosity coefficient. Based on the uniform estimates, we also obtain the convergence rate of the viscous solutions to the inviscid ones with a rate of convergence.

Here we present a multiscale method to calculate the saddle point associated with the effective dynamics arising from a stochastic system which couples slow deterministic drift and fast stochastic dynamics. This problem is motivated by the transition states on free energy surfaces in chemical physics. Our method is based on the gentlest ascent dynamics which couples the position variable and the direction variable and has the local convergence to saddle points. The dynamics of the direction vector is derived in terms of the covariance function with respective to the equilibrium distribution of the fast stochastic process. We apply the multiscale numerical methods to efficiently solve the obtained multiscale gentlest ascent dynamics, and discuss the acceleration techniques based on the adaptive idea. The examples of stochastic ordinary and partial differential equations are presented.

The basic analytical properties of the drift-diffusion-Poisson-Boltzmann system in the alternating-current (AC) regime are shown. The analysis of the AC case differs from the direct-current (DC) case and is based on extending the transport model to the frequency domain and writing the variables as periodic functions of the frequency in a small-signal approximation. We first present the DC and AC model equations to describe the three types of material in nanowire field-effect sensors: The drift-diffusion-Poisson system holds in the semiconductor, the Poisson-Boltzmann equation holds in the electrolyte, and the Poisson equation provides self-consistency. Then the AC model equations are derived. Finally, existence and local uniqueness of the solution of the AC model equations are shown. Real-world applications include nanowire field-effect bio- and gas sensors operating in the AC regime, which was only demonstrated experimentally recently. Furthermore, nanopore sensors are governed by the system of model equations and the analysis as well.

In this article we obtain Holder estimates for solutions to second-order Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the diffusion and the smoothness of the Hamiltonion. Our work is in the spirit of a result by P. Cardaliaguet and L. Silvestre.

We present a mathematical theory of time-harmonic wave propagation and reflection in a two-dimensional random acoustic waveguide with sound soft boundary and turning points. The boundary has small fluctuations on the scale of the wavelength, modeled as random. The waveguide supports multiple propagating modes. The number of these modes changes due to slow variations of the waveguide cross-section. The changes occur at turning points, where waves transition from propagating to evanescent or the other way around. We consider a regime where scattering at the random boundary has significant effect on the wave traveling from one turning point to another. This effect is described by the coupling of its components, the modes. We derive the mode coupling theory from first principles, and quantify the randomization of the wave and the transport and reflection of power in the waveguide. We show in particular that scattering at the random boundary may increase or decrease the net power transmitted through the waveguide depending on the source.

We analyze the stability of implicit-explicit flux-splitting schemes for stiff systems of conservation laws. In particular, we study the modified equation of the corresponding linearized systems. We first prove that symmetric splittings are stable, uniformly in the singular parameter ε. Then we study non-symmetric splittings. We prove that for the isentropic Euler equations, the Degond-Tang splitting [Degond & Tang, Comm. Comp. Phys. 10 (2011), pp. 1-31] and the Haack-Jin-Liu splitting [Haack, Jin & Liu, Comm. Comp. Phys. 12 (2012), pp. 955 - 980], and for the shallow water equations the recent RS-IMEX splitting are strictly stable in the sense of Majda-Pego. For the full Euler equations, we find a small instability region for a flux splitting introduced by Klein [Klein, J. Comp. Phys. 121 (1995), pp. 213-237], if this splitting is combined with an IMEX scheme as in [Noelle, Bispen, Arun, Lukacova, Munz, SIAM J. Sci. Comp. 36 (2014), pp. B989-B1024].

Markov jump processes are widely used to model natural and engineered processes. In the context of biological or chemical applications one typically refers to the chemical master equation (CME), which models the evolution of the probability mass of any copy-number combination of the interacting particles. When many interacting particles ("species") are considered, the complexity of the CME quickly increases, making direct numerical simulations impossible. This is even more problematic when one aims at controlling the Markov jump processes defined by the CME. In this work, we study both open loop and feedback optimal control problems of the Markov jump processes in the case that the controls can only be switched at fixed control stages. Based on Kurtz's limit theorems, we prove the convergence of the respective control value functions of the underlying Markov decision problem as the copy numbers of the species go to infinity. In the case of the optimal control problem on a finite time-horizon, we propose a hybrid control policy algorithm to overcome the difficulties due to the curse of dimensionality when the copy number of the involved species is large. Two numerical examples demonstrate the suitability of both the analysis and the proposed algorithms.

We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the tendency to concentrate dominates the strength of diffusion and solutions emanating from sufficiently localised initial data may explode in finite time. The main purpose of this paper is to show that under suitable spectral conditions on the flow, which guarantee good mixing properties, for any regular initial datum the solution to the corresponding advection-aggregation-diffusion equation is global if the prescribed flow is sufficiently fast. This paper can be seen as a partial extension of Kiselev and Xu (Arch. Rat. Mech. Anal. 222(2), 2016), and our arguments show in particular that the suppression mechanism for the classical 2D parabolic-elliptic Keller-Segel model devised by Kiselev and Xu also applies to the fractional Keller-Segel model (where $\triangle$ is replaced by $-(-\triangle)^\frac{\gamma}{2}$) requiring only that $\gamma >1$. In addition, we remove the restriction to dimension $d< 4$. As a by-product, a characterisation of the class of relaxation enhancing flows on the $d$-torus is extended to the case of fractional dissipation.

Recently, Jameson Cahill and Dustin G. Mixon completely characterize sensing operators in many compressed sensing instances with a robust width property. The introduced property allows uniformly stable and robust reconstruction via constrained convex optimization. However, their theory does not cover the Lasso and the Dantzig selector models, both of which are popular alternatives in statistics and optimization community. In this note, we show that the robust width property can be perfectly applied to these two models as well. Our main results solve the open problem left by Jameson Cahill and Dustin G. Mixon.

By using numerical and analytical methods, we describe the generation of fine-scale lateral electromagnetic waves, called surface plasmon-polaritons (SPPs), on atomically thick, metamaterial conducting sheets in two spatial dimensions (2D). Our computations capture the two-scale character of the total field and reveal how each edge of the sheet acts as a source of an SPP that may dominate the diffracted field. We use the finite element method to numerically implement a variational formulation for a weak discontinuity of the tangential magnetic field across a hypersurface. An adaptive, local mesh refinement strategy based on a posteriori error estimators is applied to resolve the pronounced two-scale character of wave propagation and radiation over the metamaterial sheet. We demonstrate by numerical examples how a singular geometry, e.g., sheets with sharp edges, and sharp spatial changes in the associated surface conductivity may significantly influence surface plasmons in nanophotonics.

The stochastic 3D Navier-Stokes equation with damping driven by a multiplicative noise is considered in this paper. The existence of invariant measures is proved for 3 < \beta \le 5 with any \alpha \ge 0 and \alpha \ge 1/2 as \beta=3. Using asymptotic strong Feller property, the uniqueness of invariant measures is obtained. The existence of a random attractor for the random dynamical systems generated by the solution of stochastic Navier-Stokes equation with damping is proved for \beta >3 with any \alpha >0 and \alpha \ge 1/2 as \beta=3.

We show local well-posedness of fluid-vacuum free-boundary magnetohydrodynamic(MHD) with both kinematic viscosity and magnetic diffusivity under the gravity force. We consider three-dimensional problem with finite depth and impose zero magnetic field condition on the free boundary and in vacuum. Sobolev-Slobodetskii space (Fractional Sobolev space) is used to perform energy estimates. Main difficulty is to control strong nonlinear couplings between velocity and magnetic fields. In [Lee, D., SIAM J. Math. Anal. 49, no.4, 2710-2789 (2017)], we send both kinematic viscosity and magnetic diffusivity to zero with same speed to get ideal (inviscid) free-boundary magnetohydrodynamics using the result of this paper.

In this paper, we mainly consider the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with dynamic boundary conditions and two polynomial growth nonlinearities of arbitrary order. We prove the existence of a finite dimensional global attractor for the Cahn-Hilliard-Navier-Stokes system with dynamic boundary conditions by using the l-trajectories method.

In this paper we investigate the sensitivity of the LWR model on network to its parameters and to the network itself. The quantification of sensitivity is obtained by measuring the Wasserstein distance between two LWR solutions corresponding to different inputs. To this end, we propose a numerical method to approximate the Wasserstein distance between two density distributions defined on a network. We found a large sensitivity to the traffic distribution at junctions, the network size, and the network topology.

We consider the 3D Euler equations with Coriolis force (EC) in the whole space. We show long-time solvability in Besov spaces for high speed of rotation \Omega and arbitrary initial data. For that, we obtain \Omega-uniform estimates and a blow-up criterion of BKM type in our framework. Our initial data class is larger than previous ones considered for (EC) and covers borderline cases of the regularity. The uniqueness of solutions is also discussed.

The two dimensional Primitive Equations with Levy noise are studied in this paper. A number of exponential estimates of the the solutions as well as the exponential convergence of the approximating solutions have been established, finally, a large deviation principle has been obtained.

In this article we introduce a new Riemann solver for traffic flow on networks. The Priority Riemann solver (PRS) provides a solution at junctions by taking into consideration priorities for the incoming roads and maximization of through flux. We prove existence of solutions for the solver for junctions with up to two incoming and two outgoing roads and show numerically the comparison with previous Riemann solvers. Additionally, we introduce a second version of the solver that considers the priorities as softer constraints and illustrate numerically the differences between the two solvers.