The structure of the electric double layer has long been described by the classical Poisson--Boltzmann (PB) theory, in which a uniform dielectric coefficient is often assumed. Experimental data and molecular simulations evidence that the effective dielectric coefficient decreases as local ionic concentrations. In this work, we develop a modified PB theory with concentration-dependent dielectrics that are described by the Bruggeman equation, which defines an implicit function of the effective dielectric coefficient on both the size and dielectric coefficients of hydrated ions. The Bruggeman equation, which takes into account contributions from both counterions and coions systematically, provides a closure to the modified PB theory, making the theory self-consistent. In addition to ionic size and valence, our theory introduces another source of ion-specificity, i.e., the dielectric coefficient of hydrated ions, to the continuum modeling of electrostatics. Asymptotic analysis reveals the connection between the modified PB theory and previous linear decrement models, and derives a criterion for the presence of counterion saturation. Robust numerical methods with efficient acceleration techniques are proposed to solve the resulting coupled equations. Dielectric coefficients predicted by our theory show good agreement with the experimental data for certain electrolytes. The dielectric decrement effect on the ionic structure of electric double layers is assessed in extensive numerical simulations. With ion-specific parameters, our theory predicts asymmetric camel-shape profiles of differential capacitance against applied potentials for electrolytes with low salinity, and asymmetric bell-shape profiles for electrolytes with high salinity. The impact of counterion saturation, arising from the steric effects and dielectric decrement, on the shape of differential capacitance profiles is demonstrated through analysis and numerical investigations. To further understand the effect of concentration-dependent dielectrics, the modified PB theory is also applied to study the distribution of counterions around charged cylinders with various dielectric coefficients.

In this paper, an asymptotic traveling wave of a free boundary problem related to a pricing model for corporate bond with multiple credit rating migration risk is studied. The pricing model is captured by a free boundary problem and the existence, uniqueness and regularity of the solution are obtained such that the rationality of the model guaranteed. The existence of the unique traveling wave in the free boundary problem is established with some condition of the risk discount rate satisfied. The inductive method is applied to overcome the multiplicity of free boundaries. We prove that the solution of the pricing model for corporate bond is convergent to the traveling wave, which shows a clear dynamics of price change for the corporate bond.

We are concerned with an inverse problem associated with the fractional Helmholtz system that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We are particularly interested in the case that both the medium parameter and the internal source of the wave equation are unknown. Moreover, we consider a general class of source functions which can be frequency-dependent. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.

Bending vibrations of thin beams and plates may be described by nonlinear Euler-Bernoulli beam equations with $x$-dependent coefficients. In this paper we investigate existence of families of time-periodic solutions to such a model using Lyapunov-Schmidt reduction and a differentiable Nash-Moser iteration scheme. The results hold for all parameters $(\epsilon,\omega)$ in a Cantor set with asymptotically full measure as $\epsilon\rightarrow 0$.

We prove existence of solutions to an anisotropic Cahn-Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate auxiliary results which play an essential part in the present existence proof. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1.

This paper discusses an investment problem for a single agent with higher borrowing interest rate than lending in the market. The objective is to maximize the expected discounted utility of terminal wealth by choosing portfolio of one risk asset and the bank account. The objective function is the solution of a free boundary problem with two nonlinear equations and one linear equation. The main contribution is that the existence of free boundary lines is proved in all situations and the design methods can be generally applied to other similar problem.

We translate a coagulation-framentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the \blue{nonlinear} evolution equation with a Markov jump process of a type introduced in classic work of H.~McKean. In particular this formalizes a model suggested by H.-S. Niwa [J.~Theo.~Biol.~224 (2003)] with simple coagulation and fragmentation rates. Based on the jump process, we develop a numerical scheme that allows us to approximate the equilibrium for the Niwa model, validated by comparison to analytical results by Degond et al. [J.~Nonlinear Sci.~27 (2017)], and study the population and size distributions for more complicated rates. Furthermore, the simulations are used to describe statistical properties of the underlying jump process. We additionally discuss the relation of the jump process to models expressed in stochastic differential equations and demonstrate that such a connection is justified in the case of nearest-neighbour interactions, as opposed to global interactions as in the Niwa model.

The hierarchical interpolative factorization for elliptic partial differential equations is a fast algorithm for approximate sparse matrix inversion in linear or quasilinear time. Its accuracy can degrade, however, when applied to strongly ill-conditioned problems. Here, we propose a simple modification that can significantly improve the accuracy at no additional asymptotic cost: applying a block Jacobi preconditioner before each level of skeletonization. This dramatically limits the impact of the underlying system conditioning and enables the construction of robust and highly efficient preconditioners even at quite modest compression tolerances. Numerical examples demonstrate the performance of the new approach.

In this note we propose a new approach towards solving numerically optimal stopping problems via reinforced regression based Monte Carlo algorithms. The main idea of the method is to reinforce standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimated continuation values. The proposed methodology is illustrated by several numerical examples from mathematical finance.

An important challenge in plasma physics is to determine whether ionized gases can be confined by strong magnetic fields. After properly formulating the model, this question leads to a penalized version of the Relativistic Vlasov Maxwell system, marked by the role of a singular factor corresponding to the inverse of a cyclotron frequency. In this paper, we prove in this context the existence of classical solutions for a time independent of the singular factor. We also investigate the stability of these smooth solutions.

Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential equations into the theoretical framework of diffusion approximation, extending the validity of the weak approximation from finite to infinite time horizon. The new techniques developed in this paper enable us to characterize the asymptotic behavior of constant-step-size SGD algorithms for strongly convex objective functions, a goal previously unreachable within the diffusion approximation framework. Our analysis builds upon a truncated formal power expansion of the solution of a stochastic modified equation arising from diffusion approximation, where the main technical ingredient is a uniform-in-time weak error bound controlling the long-term behavior of the expansion coefficient functions near the global minimum. We expect these new techniques to greatly expand the range of applicability of diffusion approximation to cover wider and deeper aspects of stochastic optimization algorithms in data science.

In the late stage of thin liquid films, liquid droplets are connected by an ultra thin residual film. Experimental studies and numerical simulations show that the size distributions of liquid droplets approach a self-similar form. However, theoretical study of the size distributions is lacking because it has been a challenge to retrieve statistical information from the mathematical PDE model of thin films. To facilitate the study of the statistical information, we rigorously derive a mean field model for the Ostwald ripening of thin liquid films through homogenization. This model corresponds to the dilute limit when the droplets are far away from each other and occupy a very small part of the thin film. Our analysis captures the screening effect of the droplets and shows that the mean field spatially varies in a length scale proportional to the screening length.

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. We introduce the basics of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic, and apply it to a 2x2 non conservative toy model, and a 3x3 gas dynamics system in Lagrangian coordinates. The strategy allows in particular to compute {\it exactly} isolated shocks. Numerical evidences are proposed.

We investigate efficient numerical methods for nonlinear Hamiltonian systems. Three polynomial spectral methods (including spectral Galerkin, Petrov-Galerkin, and collocation methods) coupled with domain decomposition are presented and analyzed. Our main results include the energy and symplectic structure-preserving properties and error estimates. We prove that the spectral Petrov-Galerkin method preserves the energy exactly while both the spectral Gauss collocation and spectral Galerkin methods are energy conserving up to spectral accuracy. While it is well known that collocation at Gauss points preserves symplectic structure, we prove that the Petrov-Galerkin method preserves the symplectic structure up to a Gauss numerical quadrature error and the spectral Galerkin method preserves the symplectic structure up to spectral accuracy error. Finally, we show that all three methods converge exponentially, which makes it possible to simulate the long time behavior of the system. Numerical experiments indicate that our algorithms are efficient.

In this paper we study Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mb{R}^n$, $n\geq1$, where the heat conduction is modeled by Fourier's law. We explain some qualitative properties of solutions influenced by different damping mechanisms. We show which damping in the model has a dominant influence on smoothing effect, energy estimates, $L^p-L^q$ estimates not necessary on the conjugate line, and on diffusion phenomena. Moreover, we derive asymptotic profiles of solutions in a framework of weighted $L^1$ data. In particular, sharp decay estimates for lower bound and upper bound of solutions in the $\dot{H}^s$ norm ($s\geq0$) are shown.

We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density $u$. In case of \emph{fast-decay} mobilities, namely mobilities functions under a Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density $\rho$ is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density $\rho$ allow us to motivate the aforementioned change of variable and to state the results in terms of the original density $u$ without prescribing any boundary conditions.

The proposal of this paper is to study the well-posedness and properties of solutions to the boundary layer problem for wind-driven oceanic current, which differs from the classical Prandtl boundary layer equations with a nonlocal integral term arising from the Coriolis force. First, under Oleinik's monotonic condition \cite{O} on the tangential velocity field, we obtain the local well-posedness of the boundary layer problem by using the Crocco transformation. Secondly, we show that the back flow point appears at the physical boundary in a finite time under certain constraint on the growth rate of the tangential velocity when both of the initial tangential velocity and the upstream velocity are monotonically increasing with respect to the normal variable of the boundary, even if the momentum of the outer flow is favorite for the classical Prandtl equations, in the sense with this favorite condition there will be no back flow in the two-dimensional Prandtl boundary layer. This shows that the factor of the Coriolis force stimulates the appearance of the back flow of the boundary layer.

The connection between forward backward doubly stochastic differential equations and the optimal filtering problem is established without using the Zakai equation. The solutions of forward backward doubly stochastic differential equations are expressed in terms of conditional law of a partially observed Markov diffusion process. It then follows that the adjoint time-inverse forward backward doubly stochastic differential equations governs the evolution of the unnormalized filtering density in the optimal filtering problem.

In the study of dynamical and physical systems, the input parameters are often uncertain or randomly distributed according to a measure $\varrho$. The system's response $f$ pushes forward $\varrho$ to a new measure $f\circ \varrho$ which we would like to study. However, we might not have access to $f$, but to its approximation $g$. We thus arrive at a fundamental question -- if $f$ and $g$ are close in $L^q$, does the measure $g\circ \varrho$ approximate $f\circ \varrho$ well, and in what sense? In scientific computing, these settings are common in surrogate models for uncertainty quantification (UQ), where the convergence of the measures has until recently remained an open question. Previously, it was demonstrated that the answer to this question might be negative in terms of the $L^p$ distance between probability density functions (PDF). Instead, we show in this paper that the Wasserstein metric is the proper framework for this question. We bound the Wasserstein distance~$W_p (f\circ \varrho , g\circ \varrho) $ from above by $\|f-g\|_{q}$, and provide lower bounds for the cases where $p=1,2$. From a numerical analysis standpoint, since the Wasserstein distance is related to the cumulative distribution functions (CDF), we show that the latter is well approximated by methods such as spline interpolation and generalized polynomial chaos (gPC).

We introduce an asymptotic solution form, termed as extended Wentzel-Kramers-Brillouin (E-WKB), to solve the high-frequency vectorial wave equation when the initial Cauchy data is prescribed as the traditional Wentzel-Kramers-Brillouin (WKB) function. The E-WKB form, formulated as an integral of a family of Gaussian coherent states, can be regarded as an extension of the WKB form. The domain of the integral is the Lagrangian submanifold induced by the underlying Hamiltonian flow. Although the procedure of solving wave equations by using the E-WKB form is parallel to that of the classical WKB analysis, the former can overcome the difficulty due to the presence of caustic points. We present numerical tests on the time harmonic acoustic wave equation and the Helmholtz equation to validate the proposed asymptotic theory.

The dual continuum model serves as a powerful tool in the modeling of subsurface applications. It allows a systematic coupling of various components of the solutions. The system is of multiscale nature as it involves high heterogeneous and high contrast coefficients. To numerically compute the solutions, some types of reduced order methods are necessary. We will develop and analyze a novel multiscale method based on the recent advances in multiscale finite element methods. Our method will compute multiple local multiscale basis functions per coarse region. The idea is based on some local spectral problems, which are important to identify high contrast channels, and an energy minimization principle. Using these concepts, we show that the basis functions are localized, even in the presence of high contrast long channels and fractures. In addition, we show that the convergence of the method depends only on the coarse mesh size. Finally, we present several numerical tests to show the performance.

In this work we propose a novel strategy to define high-order fully well-balanced Lagrange-Projection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system as it is a reference system of balance laws with non-trivial stationary solutions. Nevertheless, the strategy proposed here could be extended to other interesting balance laws. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions. Following Castro et al. 2017, we exploit the idea of using a high-order well-balanced reconstruction operator for the Lagrangian step. Nevertheless, this is not enough to achieve well-balanced high-order during the projection step. We propose here a new projection step that overcomes this difficulty and that reduces to the standard one in case of conservation laws. Finally, some numerical experiments illustrate the good behaviour of the scheme.

In this paper, we study the global existence and extensibility criterion of large-amplitude solutions to a chemotaxis-fluid model in bounded domains. The model under consideration is an integrated version of several recently studied models in bio-fluids, which is a coupled system of partial differential equations with strong nonlinearities. The results obtained in this paper appear to be among the first ones for the integrated model.

In the present contribution, we investigate first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and non-conservative first order terms. Whereas the theory of first-order systems of conservation laws is well established and the conditions for the existence of a supplementary conservation law for smooth solutions, well known, there exists so far no general extension when non-conservative terms are present. We propose a framework in order to extend the existing theory and show that the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to a supplementary conservation law. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms to obtain a supplementary conservation law. The theory as well as developed computer algebra tool are then applied to a Baer-Nunziato two-phase flow model and to a multicomponent plasma fluid model. The first one is a first-order fluid model, with non-conservative terms impacting on the linearly degenerate field and requires a closure since there is no way to derive interfacial quantities from averaging principles and we need guidance in order to close the pressure and velocity of the interface and the thermodynamics of the mixture. The second one involves first order terms for the heavy species coupled to second order terms for the electrons, the non-conservative terms impact the genuinely nonlinear fields and the model can be rigorously derived from kinetic theory. We show how the theory allows to recover the whole spectrum of closures obtained so far in the literature for the two-phase flow system as well as conditions when one aims at extending the thermodynamics and also applies to the plasma case, where we recover the usual entropy supplementary equation, thus assessing the effectiveness and scope of the proposed theory.

We consider the Cauchy problem to the 1D non-resistive compressible magnetohydrodynamics (MHD) equations. We established the global existence and uniqueness of strong solutions for large initial data and vacuum when the viscosity coefficient is assumed to be constant or density-dependent. The analysis is based on the full use of effective viscous flux and the Caffarelli-Kohn-Nirenberg weighted inequality to get the higher-order estimates of the solutions. This result could be viewed as the first one on the global well-posedness of strong solutions to the Cauchy problem of 1D non-resistive compressible MHD equations while the initial data may be arbitrarily large and permit vacuum.

In this paper, we investigate the smooth solutions of the antiferromagnets Landau-Lifshitz-Bloch(LLB) equation, which can describe the dynamics of micromagnetic under high temperature. The existence and uniqueness of smooth solutions for LLB equation in R^{2} and R^{3} are proved.

The current paper is devoted to the investigation of the global-in-time stability of large solutions to the compressible liquid crystal equations in the whole space. Suppose that the density is bounded from above uniformly in time in the Hoder space $C^\alpha$ with $\alpha$ sufficiently small and in $L^\infty$ space respectively. Then we prove two results: (1). Such kind of the solution will converge to its associated equilibrium with a rate which is the same as that for the heat equation. (2). Such kind of the solution is stable, which means any perturbed solution will remain close to the reference solution if initially they are close to each other. This implies that the set of the smooth and bounded solutions is open.

We study the Cauchy problem of density-dependent magnetic Benard system with zero density at infinity on the whole two-dimensional space. We prove that there admits a unique local strong solution provided the initial density and the initial magnetic decay not too slow at infinity. In particular, there is no need to require any Choe-Kim type compatibility condition for the initial data.

In this paper, two approaches for modeling three-component fluid flows using diffusive interface method are discussed. Thermodynamic consistency of the proposed models is preserved when using an energetic variational framework to derive the coupled systems of partial differential equations that comprise the resulting models. The issue of algebraic and dynamic consistency is investigated. In addition, the two approaches that are presented are compared analytically and numerically.

In this paper, we consider the Cauchy problem of the modified KdV(mKdV) equation. Local well-posedness and ill-posedness of this problem are obtained in modulation spaces. our results contains all of the subcritical data in modulation spaces.

We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]), we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant heat conductivity case to the degenerate and nonlinear one under stress-free and thermally insulated boundary conditions.

This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-\Delta)^\alpha u$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $\alpha=1$. This paper discovers that there are new phenomena with the case $\alpha< 1$. The approach for $\alpha=1$ can not be directly extended to $\alpha< 1$. We establish that, for $\alpha< 1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^\sigma_{2,\infty}(\mathbb R^d)$ with $\sigma > 1+\frac{d}{2}-\alpha$ leads to a unique local solution. For the case $\alpha\ge 1$, $u_0$ in the homogeneous Besov space $\mathring B^{1+\frac{d}{2}-2\alpha}_{2,1}(\mathbb R^d)$ and $b_0$ in $ \mathring B^{1+\frac{d}{2}-\alpha}_{2,1}(\mathbb R^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal.

Solutions of first-order nonlinear hyperbolic conservation laws typically develop shocks in finite time even with smooth initial conditions. However, in heterogeneous media with rapid spatial variation, shock formation may be delayed or avoided because the variation in the medium introduces an effective dispersion. When shocks do form in such media, their speed of propagation depends in a non-trivial way on the material structure. We investigate conditions for shock formation and propagation in heterogeneous media. We focus on the propagation of plane waves in two-dimensional periodic media with material variation in only one direction. We propose an estimate for the speed of the shocks that is based on the Rankine-Hugoniot conditions applied to a leading-order homogenized (constant coefficient) system. We verify this estimate via numerical simulations using different nonlinear constitutive relations and layered and smoothly varying periodic media. In addition, we discuss conditions and regimes under which shocks form in this type of media.

Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main challenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nilpotent Lie algebras which are called the left and right nilpotent algebras of the solvable algebra. The right nilpotent algebra is the smallest ideal of the lower central series of the solvable algebra, while the left nilpotent algebra is the factor algebra of the solvable algebra and its right nilpotent algebra. We show that the solvable algebras are decomposable if its left nilpotent algebra is an Abelian algebra of dimension higher than one and its right algebra is an Abelian algebra of dimension one. We further show that all the solvable algebras are isomorphic if their left nilpotent algebras are Heisenberg algebras of fixed dimension and their right algebras are Abelian algebras of dimension one.

This paper is concerned with a coupled Navier-Stokes/Cahn-Hilliard system describing a diffuse interface model for the two-phase flow of compressible viscous fluids in a bounded domain in one dimension. In the case of concentration dependent viscous coefficient, we obtain the local existence of unique strong solution for $\rho_0\in H^3(I)$. In the case of constant viscous coefficient, we prove the global existence of classical, weak and strong solutions for $\rho_0\in C^{3, \alpha}(I)$, $\rho_0\in H^1(I)$ and $\rho_0\in H^2(I)$, respectively. Moreover, we also give a more general uniqueness result such that the uniqueness of classical and strong solutions be an immediate consequence. In this paper, we assume that the initial density function $\rho_0$ has a positive lower bound.

In this paper, we consider the existence of global weak solutions to a one-dimensional fluid-particles interaction model: inviscid Burgers-Vlasov equations with fluid velocity in $L^\infty$ and particles' probability density in $L^1$. Our weak solution is also an entropy solution to the inviscid Burgers' equation. The approach is adding ingeniously artificial viscosity to construct approximate solutions satisfying $L^\infty$ compensated compactness framework and weak $L^1$ compactness framework. It is worthy to be pointed out that the bounds of fluid velocity and the kinetic energy of particles' probability density are both independent of time.

It is well known that conjugate gradient methods are suitable for large-scale nonlinear optimization problems, due to their simple calculation and low storage. In this paper, we present a three-term conjugate gradient method using subspace technique for large-scale unconstrained optimization, in which the search direction is determined by minimizing the quadratic approximation of the objective function in the subspace spanned by the vectors: $-g_{k+1}$, $s_k$ and $y_k$. We show the search direction can both satisfy the descent condition and Dai-Liao conjugacy condition. Under proper assumptions, global convergence result of the proposed method is established. Numerical experiments show the proposed method is efficient and robust.

It is well known that conjugate gradient methods are suitable for large-scale nonlinear optimization problems, due to their simple calculation and low storage. In this paper, we present a three-term conjugate gradient method using subspace technique for large-scale unconstrained optimization, in which the search direction is determined by minimizing the quadratic approximation of the objective function in the subspace spanned by the vectors: $-g_{k+1}$, $s_k$ and $y_k$. We show the search direction can both satisfy the descent condition and Dai-Liao conjugacy condition. Under proper assumptions, global convergence result of the proposed method is established. Numerical experiments show the proposed method is efficient and robust.

We propose a diagram notation for the derivation of hyperbolic moment models for the Boltzmann equation that yields a better understanding of the resulting moment systems. So far several hyperbolic moment models were presented, but their derivations are often very technical and there is little insight into the explicit form of the equations. In our diagram notation, each term in the moment equations can be explicitly tracked throughout the derivation process and whether the resulting moment system is hyperbolic can be easily observed from the diagram. We apply the diagram notation to derive existing moment models, including Grad's moment equations, hyperbolic moment equations, quadrature-based moment equations, and explain differences among them. Due to its explicit nature, the diagram notation succeeds in the straightforward derivation and understanding of a new, largely reduced model called Simplified Hyperbolic Moment Equations (SHME). Numerical results of a 1D shock tube show that the simplifications are too strong to obtain convergence for a large number of moments but the simplified model gives accurate solutions for small number of moments.

We propose fully discrete, implicit-in-time finite volume schemes for general nonlinear nonlocal Fokker-Planck type equations with a gradient flow structure, usually referred to as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preserving and energy-decaying properties, essential for their practical use. The first order in time and space scheme unconditionally verifies these properties for general nonlinear diffusion and interaction potentials while the second order scheme does so provided a CFL condition holds. Dimensional splitting allows for the construction of these schemes with the same properties and a reduced computational cost in higher dimensions. Numerical experiments validate the schemes and show their ability to handle complicated phenomena in aggregation-diffusion equations such as free boundaries, metastability, merging and phase transitions.

This paper concerns with the stability of the plane Couette flow resulted from the motions of boundaries that the top boundary $\Sigma_1$ and the bottom one $\Sigma_0$ move with constant velocities $(a,0)$ and $(b,0)$, respectively. If one imposes Dirichlet boundary condition on the top boundary and Navier boundary condition on the bottom boundary with Navier coefficient $\alpha$, there always exists a plane Couette flow which is exponentially stable for nonnegative $\alpha$ and any positive viscosity $\mu$ and any $a, b \in \mathbb{R}$, or, for $\alpha< 0$ but viscosity $\mu$ and the moving velocities of boundaries $(a,0), (b,0)$ satisfy some conditions, see Theorem 1.1. However, if we impose Navier boundary conditions on both boundaries with Navier coefficients $\alpha_0$ and $\alpha_1$, then it is proved that there also exists a plane Couette flow (including constant flow or trivial flow) which is exponentially stable provided that any one of two conditions on $\alpha_0,\alpha_1$, $a, b$ and $\mu$ in Theorem 1.2 holds. Therefore, the known results on the stability of incompressible Couette flow to Dirichlet boundary value problems are extended to the Navier boundary value problems.

Consideration in the present paper is a mathematical model proposed as an equation of long-crested shallow-water waves propagating in one direction with the effect of Earth's rotation. The system is called Rotation-Camassa-Holm system(RCH2). The local well-posedness of the periodic Cauchy problem is then established by the linear transport theory. Then, wave-breaking phenomena is investigated based on the method of characteristics and the Riccati-type differential inequality with two different kinds of method. Finally, the waving breaking data are illustrated and the existence of global solutions is obtained in details for the periodic RCH2 system.

t is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_n=2\cos\l(\binom{n}{2}\omega +ny+x\r)$ with $\omega$ an irrational number. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when $\omega$ is the golden mean, which allows to derive the positivity of the infinite-volume Lyapunov exponent from three conditions imposed at a fixed, finite scale. Here we numerically verify the two conditions among these that are amenable to computer calculations.