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.

The capability of using imperfect stochastic reduced-order models to capture crucial passive tracer statistics is investigated. The passive scalar field is advected by a two-layer baroclinic turbulent flow which can generate various representative regimes in atmosphere and ocean. Much simpler and more tractable linear Gaussian stochastic models are proposed to approximate the complex and high-dimensional advection flow equations. The imperfect model prediction skill is improved through a judicious calibration of the model errors using leading order statistics of the background advection flow, while no additional prior information about the passive tracer field is required. A systematic framework of correcting model errors with empirical information theory is introduced, and optimal model parameters under this unbiased information measure can be achieved in a training phase before the prediction. It is demonstrated that crucial principal statistical quantities like the tracer spectrum and fat-tails in the tracer probability density functions in the most important large scales can be captured efficiently with accuracy using the reduced-order tracer model in various dynamical regimes of the flow field with distinct statistical structures. The skillful linear Gaussian stochastic modeling algorithm developed here should also be useful for other applications such as accurate forecast of mean responses and efficient algorithms for state estimation or data assimilation.

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.

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.

New fractional r-order seminorms, TGV^r, r \in R, r \ge 1, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order TGV^k-seminorms, k \in N. The fractional r-order TGV^r-seminorms are shown to be intermediate between the integer order TGV^k-seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect to parameters and order of derivation is performed. Existence of solutions to the bilevel training scheme is proved by \Gamma-convergence. Finally, the numerical landscape of the cost function associated to the bilevel training scheme is discussed for two numerical examples.

In this paper we study the dynamics of vesicle membranes in incompressible viscous fluids. We prove existence and uniqueness of the local strong solution for this model coupling of the Navier-Stokes equations with a phase field equation in an L_p-P_q setting. We transform the equation into a quasi-linear parabolic evolution equation and use the general theory proved by Pruss et al.. Since the operator and the nonlinear term are analytic, we have that the solution is real analytic in time and space. At last it is shown that the variational strict stable solution is exponentially stable, provided the product of the viscosity coefficient and the mobility constant is large.

The Poisson--Nernst--Planck (PNP) equations have been widely applied to describe ionic transport in ion channels, nanofluidic devices, and many electrochemical systems. Despite their wide applications, the PNP equations fail in predicting dynamics and equilibrium states of ionic concentrations in confined environments, due to the ignorance of the excluded volume effect. In this work, a simple but effective modified PNP (MPNP) model with the excluded volume effect is derived, based on a modification of diffusion coefficients of ions. At the steady state, a modified Poisson--Boltzmann (MPB) equation is obtained with the help of the Lambert-W special function. The existence and uniqueness of a weak solution to the MPB equation are established. Further analysis on the limit of weak and strong electrostatic potential leads to two modified Debye screening lengths, respectively. A numerical scheme that conserves total ionic concentration and satisfies energy dissipation is developed for the MPNP model. Numerical analysis is performed to prove that our scheme respects ionic mass conservation and satisfies a corresponding discrete free energy dissipation law. Positivity of numerical solutions is also discussed and numerically investigated. Numerical tests are conducted to demonstrate that the scheme is of second-order accurate in spatial discretization and has expected properties. Extensive numerical simulations reveal that the excluded volume effect has pronounced impacts on the dynamics of ionic concentration and flux. In addition, the effect of volume exclusion on the timescales of charge diffusion is systematically investigated by studying the evolution of free energies and diffuse charges.

We study the solution of a non-Newtonian flow in a porous medium which characteristic size of the pores $\varepsilon$ and containing a fissure of width $\eta_\varepsilon$. The flow is described by the incompressible Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index $1< p< 2$ (pseudoplastic fluids). We consider the limit when size of the pores tends to zero and we obtain different models depending on the magnitude $\eta_{\varepsilon}$ with respect to $\varepsilon$.

In this paper, we study the quasi-neutral limit of the full quantum Navier-Stokes-Maxwell equation as the Debye length tends to zero. We justify rigorously the quasi-neutral limit by establishing rigorous uniform estimates on the error functions with respect to the Debye length and by using the formal asymptotic expansion and singular perturbation methods combined with curl-div decomposition of the gradient. The key difficulty is to deal with the quantum effect, which do play important roles in establishing a priori estimates.

We address the far field regularity for solutions of the surface quasi-geostrophic equation \begin{align*} &\theta_t+u\cdot\nabla\theta+\Lambda^{2\alpha}\theta=0 \\ &u=\RR^{\perp}\theta=(-\RR_2\theta,\RR_1\theta), \end{align*} in the supercritical range $0< \alpha< 1/2$ with $\alpha$ sufficiently close to $1/2$. We prove that if the datum is sufficiently regular, then the set of space-time singularities is compact in ${\mathbb R}^2\times{\mathbb R}$. The proof depends on a new spatial decay result on solutions in the supercritical range.

The 3D Boussinesq equations are one of the most important models for geophysical fluids. The fundamental problem of whether or not reasonably smooth solutions to the 3D Boussinesq equations with the standard Laplacian dissipation can blow up in a finite time is an outstanding open problem. The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations, but also are physically relevant and mathematically important. This paper focuses on a system of the 3D Boussinesq equations with fractional partial dissipation and proves that any $H^1$-initial data always leads to a unique and global-in-time solution. The result of this paper is part of our efforts devoted to the global well-posedness problem on the Boussinesq equations with minimal dissipation.

The global existence of classical solutions to reaction-diffusion systems in dimensions one and two is proved. The considered systems are assumed to satisfy an {\it entropy inequality} and have nonlinearities with at most cubic growth in 1D or at most quadratic growth in 2D. This global existence was already proved in [T. Goudon and A. Vasseur, Ann. Sci. \'Ec. Norm. Sup\'er. (4) 43 (2010), no. 1, 117--142] by a De Giorgi method. In this paper, we give a simplified proof by using a modified Gagliardo-Nirenberg inequality and the regularity of the heat operator. Moreover, the classical solution is proved to have $L^{\infty}$-norm growing at most polynomially in time. As an application, the solutions to chemical reaction-diffusion systems satisfying the so-called complex balance condition are proved to converge exponentially to equilibrium in $L^{\infty}$-norm.

In this paper, we will introduce a mathematical model of jerk equation to simulate the unstable oscillations of the motion of a falling sphere in the wormlike micellar solution.This differential/algebraic equation (DAE) is established only by learning the experimental data of time vs velocity with the sparse optimization method. To simulate the solutions of the DAE, four discretization schemes are proposed and compared. Periodic and damped harmonic motion,and nonuniform transient and sustaining oscillations can be observed for the sedimentation of a sphere through the non-Newtonian fluid in the numerical experiments.

In this paper, we provide a complete description of the selected spreading speed of systems of reaction- diffusion equations with unilateral coupling and prove the existence of anomalous spreading speeds for systems with monostable nonlinearities. Our work extends known results for systems with linear and quadratic couplings, and Fisher-KPP type nonlinearities. Our proofs rely on the construction of appropriate sub- and super-solutions.

We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.

This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly non-globally monotone nonlinearities such as stochastic Kuramoto-Sivashinsky equations. The main result of this article proves that the proposed approximation scheme converges strongly and numerically weakly to the solution process of such an SPDE. Key ingredients in the proof of our convergence result are a suitable generalized coercivity-type condition, the specific design of the accelerated exponential Euler-type approximation scheme, and an application of Fernique's theorem.

Dislocations are the main carriers of the permanent deformation of crystals. For simulations of engineering applications, continuum models where material microstructures are represented by continuous density distributions of dislocations are preferred. It is challenging to capture in the continuum model the short-range dislocation interactions, which vanish after the standard averaging procedure from discrete dislocation models. In this study, we consider systems of parallel straight dislocation walls and develop continuum descriptions for the short-range interactions of dislocations by using asymptotic analysis. The obtained continuum short-range interaction formulas are incorporated in the continuum model for dislocation dynamics based on a pair of dislocation density potential functions that represent continuous distributions of dislocations. This derived continuum model is able to describe the anisotropic dislocation interaction and motion. Mathematically, these short-range interaction terms ensure strong stability property of the continuum model that is possessed by the discrete dislocation dynamics model. The derived continuum model is validated by comparisons with the discrete dislocation dynamical simulation results.

In this work we prove local and global well-posedness of the Cauchy problem of the regularized intermediate long-wave (rILW) equation in periodic and nonperiodic Sobolev spaces.

We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker--Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached.

We consider the Helmholtz equation with a complex attenuation coefficient on a bounded, strictly convex domain in R^d. We prove a Holder conditional stability estimate for identifying attenuation coefficients from phaseless boundary value measurements, when the initial excitation state is in the form of a Gaussian bump. We use the Gaussian beam Ansatz and stability results for the X-ray transform on strictly convex domains to establish these estimates.

We study the global in time classical solutions to the two-fluid incompressible Navier-Stokes-Maxwell system with (solenoidal) Ohm's law with small initial data. This system is a coupling of the incompressible Navier-Stokes equations with the Maxwell equations through the Lorenz force and Ohm's law for the current. Comparing to the previous results, we employ the decay properties of both the electric field $E$ and the wave equation with linear damping of the divergence free magnetic field $B$.