In this paper, we analyze the behavior of viscous shock profiles of one-dimensional compressible Navier–Stokes equations with a singular pressure law which encodes the effects of congestion. As the intensity of the singular pressure tends to $0$, we show the convergence of these profiles towards free-congested traveling front solutions of a two-phase compressible-incompressible Navier–Stokes system

In this paper we study the asymptotic behavior of Brownian motion in both comb-shaped planar domains, and comb-shaped graphs. We show convergence to a limiting process when both the spacing between the teeth and the width of the teeth vanish at the same rate. The limiting process exhibits an anomalous diffusive behavior and can be described as a Brownian motion time-changed by the local time of an

In this paper we derive a Bismut–Elworthy–Li–type formula with respect to strong solutions to singular stochastic differential equations (SDE’s) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H \lt 1/2$. “Singular” here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this

In this paper, the local well-posedness of strong solutions to the Cauchy problem of the isentropic compressible Navier–Stokes equations is proved with the initial data being allowed to have vacuum. The main contribution of this paper is that the well-posedness is established without assuming any compatibility condition on the initial data, which was widely used before in many works concerning the

We study a class of fourth-order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type\[\int_{\Omega} u^{2 \gamma - \alpha - \beta} \Delta u^\alpha \Delta u^\beta dx \geq c \int_{\Omega} {\lvert \Delta u^\gamma \rvert}^2 dx \; \textrm{,}\]which

In this work, we investigate a variational formulation for a time-fractional Fokke–Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [R. Jordan, D. Kinderlehrer, and F. Otto, SIAM

In this paper, we study the long-time behavior of the solutions for the initial-boundary value problem to a one-dimensional Navier–Stokes equations for a reacting mixture in a half line $\mathbb{R}_{+} := (0, \infty)$. We give the asymptotic stability of not only stationary solution for the impermeability problem but also the composite waves consisting of the subsonic BL-solution, the contact wave

In this paper, we investigate the well-posedness/ill-posedness of the stationary solutions to the isothermal bipolar hydrodynamic model of semiconductors driven by Euler–Poisson equations. Here, the density of electrons is proposed with sonic boundary and considered in interiorly subsonic case or interiorly supersonic case, while the density of holes is considered in fully subsonic case or fully supersonic

The free boundary problem of one-dimensional heat conducting compressible Navier–Stokes equations with large initial data is investigated. We obtain the global existence of strong solution under stress-free boundary condition along the free surface, where the heat conductivity depends on temperature $(\kappa = \overline{\kappa} \theta^b , b \in (0, \infty))$ and the viscosity coefficient depends on

In this paper, we study some properties of positive solutions of nonlinear integral equations with axis-symmetric kernels, which arise from weak-type convolution-Young’s inequality and the stationary magnetic compressible fluid stars. With the help of the method of moving planes and regularity lifting lemma, we show that all of the positive solutions in certain functional spaces are symmetric and monotonically

The derivation of different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and the nonlinear progressive wave equation (NPE) from an isentropic Navier–Stokes/Euler system is systematized using the Hilbert-type expansion in the corresponding perturbative and (for the KZK and NPE equations) paraxial ansatz. The use of

Cahn–Hilliard/Navier–Stokes system is the combination of the Cahn–Hilliard equation with the Navier–Stokes equations. It describes the motion of unsteady mixing fluids and has a wide range of applications ranging from turbulent two-phase flows to microfluidics. In this paper we consider the non-local Cahn–Hilliard equation (the gradient term of the order parameter in the free energy is replaced with

In this paper, we develop a robust and efficient numerical method for shallow water equations with moving bottom topography. The model consists of the Saint-Venant system governing the water flow coupled with the Exner equation for the sediment transport. One of the main difficulties in designing good numerical methods for such models is related to the fact that the speed of water surface gravity waves

The present paper is dedicated to the study of the Cauchy problem for compressible Navier–Stokes–Poisson–Korteweg model in any dimension $d \geq 2$, which simultaneously involves the lower order potential term and the higher order capillarity term. The unique global solvability of the system is obtained when the initial data are close to a stable equilibrium state in a functional setting invariant

The Cauchy problem of the 1D compressible radiative and reactive gas without viscosity is studied in this paper. When the radiation effect is under consideration, the equations present high nonlinearity, together with the lack of viscosity, which result in many more difficulties. When the solution to the corresponding Riemann problem of the Euler equation consists of a contact discontinuity and rarefaction

The paper proves existence of renormalized solutions for a class of velocity-discrete coplanar stationary Boltzmann equations with given indata. The proof is based on the construction of a sequence of approximations with $L^1$ compactness for the integrated collision frequency and gain term. $L^1$ compactness of a sequence of approximations is obtained using the Kolmogorov–Riesz theorem and replaces

This paper is devoted to studying the inflow problem for an ideal polytropic model with non-viscous gas in one-dimensional half space. We show the existence of the boundary layer in different areas. By employing the energy method, we also prove the unique global-in-time existence of the solution and the asymptotic stability of both the boundary layers, the $3$‑rarefaction wave and their superposition

In this paper, we derive some new lower bounds of possible blow up solutions in $\dot{H}^1_p (\mathbb{R}^3)$ with $3 / 2 \lt p \lt \infty$ to the 3D Navier–Stokes equations, which provides a new proof of the corresponding recent results involving blow up rates in $\dot{H}^s$ with $1 \leq s \lt 5/2$ in [A. Cheskidov and M. Zaya, J. Math. Phys., 57:023101, 2016; J.C. Cortissoz and J.A. Montero, J. Math

We examine challenges to sampling from Boltzmann distributions associated with multiscale energy landscapes. The multiscale features—or “roughness”—correspond to highly oscillatory, but bounded, perturbations of a smooth landscape. Through a combination of numerical experiments and analysis we demonstrate that the performance of Metropolis adjusted Langevin algorithm can be severely attenuated as the

In this work, we introduce a new Monte Carlo method for solving the Boltzmann model of rarefied gas dynamics. The method works by reformulating the original problem through a micro-macro decomposition and successively by solving a suitable equation for the perturbation from the local thermodynamic equilibrium. This equation is then discretized by using unconditionally stable exponential schemes in

We propose a novel problem formulation of continuous-time information propagation on heterogeneous networks based on jump stochastic differential equations (JSDE). The structure of the network and activation rates between nodes are naturally taken into account in the JSDE. This new formulation allows for efficient and stable algorithms for a variety of challenging information propagation problems,

We show the vanishing viscosity limit to entropy shocks for the fractal Burgers equation in one space dimension. More precisely, we quantify the rate of convergence of the inviscid limit in $L^2$ for large initial perturbations around the entropy shock on any bounded time interval. This is the first result on the inviscid limit to entropy shock for the fractal Burgers equation with the quantified convergence

We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas in particular. We assume the discontinuity in the piecewise-smooth solution is an extremal shock. We use only mild hypotheses on the system. Our techniques and result

Multi-agent systems can be successfully described by kinetic models, which allow one to explore the large scale aggregate trends resulting from elementary microscopic interactions. The latter may be formalised as collision-like rules, in the spirit of the classical kinetic approach in gas dynamics, but also as Markov jump processes, which assume that every agent is stimulated by the other agents to

We introduce a model dealing with conservation laws on networks and coupled boundary conditions at the junctions. In particular, we introduce buffers of fixed arbitrary size and time-dependent split ratios at the junctions, which represent how traffic is routed through the network, while guaranteeing spill-back phenomena at nodes. Having defined the dynamics at the level of conservation laws, we lift

We present three examples of chemical reaction networks whose ordinary differential equation scaling limits are almost identical and in all cases stable. Nevertheless, the Markov jump processes associated to these reaction networks display the full range of behaviors: one is stable (positive recurrent), one is unstable (transient) and one is marginally stable (null recurrent). We study these differences

In this work we study the nonlocal transport equation derived recently by Steinerberger [Proc. Amer. Math. Soc., 147(11):4733–4744, 2019]. When this equation is considered on the real line, it describes how the distribution of roots of a polynomial behaves under iterated differentiation of the function. This equation can also be seen as a nonlocal fast diffusion equation. In particular, we study the

This paper studies the asymptotic behavior of the solution to the initial boundary value problem of a one-dimensional compressible viscous heat-conducting gas with radiation. We consider an outflow problem, where the gas blows out the region through the boundary, of the general gases including ideal polytropic gas. First, we give the necessary and sufficient conditions for an existence of the non-degenerate

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0 \lt \alpha \leq 2$. Special cases are the two-dimensional incompressible Euler equations $(\alpha = 2)$ and the surface quasi-geostrophic (SQG) equations $(\alpha = 1)$. We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when

In this note, we study the Nernst–Planck–Navier–Stokes system for the transport and diffusion of ions in electrolyte solutions. The key feature is to establish three energy-dissipation equalities. As their direct consequence, we obtain global existence for two-ionic species case in $\mathbb{R}^n , n \geq 2$, and multi-ionic species case in $\mathbb{R}^n , n=2,3$.

Sharp bounds for the Rademacher complexity and the generalization error are derived for the residual network model. The Rademacher complexity bound has no explicit dependency on the depth of the network, while the generalization bounds are comparable to the Monte Carlo error rates, suggesting that they are nearly optimal in the high dimensional setting. These estimates are achieved by constraining

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 this paper, we consider the steady irrotational Euler flows in multidimensional nozzles. The first rigorous proof on the existence and uniqueness of the incompressible flow is provided. Then, we justify the corresponding low Mach number limit, which is the first result of the low Mach number limit on the steady Euler flows. We establish several uniform estimates, which does not depend on the Mach

In this paper, we study the propagation of the mono-kinetic distribution in the Cucker–Smale-type kinetic equations. More precisely, if the initial distribution is a Dirac mass for the variables other than the spatial variable, then we prove that this “mono-kinetic” structure propagates in time. For that, we first obtain the stability estimate of measure-valued solutions to the kinetic equation, by

This paper concerns with the stability of the plane Couette flow resulting from the motions of boundaries such 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

We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker–Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical use. The first-order scheme verifies these properties

It 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: $v_n = 2 \cos ((\frac{n}{2}) \omega + ny + x)$ with $\omega$ an irrational number and $x, y \in [0, 1]$. Recently, Han, Schlag, and the second author

Under 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

In this paper, we consider the initial boundary value problem for the one-dimensional micropolar fluids for viscous compressible and heat-conducting fluids in a bounded domain with the Neumann/Robin boundary conditions on temperature. There are few results until now about global existence of regular solutions to the equations of hydrodynamics with the Robin boundary conditions on temperature. By the

We consider the quantum Zakharov system in spatial dimensions greater than $1$. The local well-posedness is obtained for initial data of the electric field and of the ion density lying in some Sobolev spaces with certain regularities. For higher dimensions, the results cover the subcritical region. We get major part of the subcritical region for lower dimensions. For the quantum Zakharov system with

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller–Segel equation with a fractional dissipation and an additional mixing effect of advection by an incompressible flow. Under a suitable mixing condition on the advection, we study well-posedness of solution with large initial data. We establish the global $L^\infty$ estimate of the solution through nonlinear maximum

We present a systematic derivation of thermodynamically consistent hydrodynamic models for multi-component, compressible viscous fluid mixtures under isothermal conditions using the generalized Onsager principle and the one-fluid multi-component formulation. By maintaining momentum conservation while enforcing mass conservation at different levels, we obtain two compressible models. When the fluid

In this paper, the asymptotic behavior of the solutions for compressible Euler system with a non-local interaction term is studied. Using velocity damping to restrain the singularity caused by the anisotropic interaction between individuals, the exponential decay estimate of the solutions is obtained.

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 Höder 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

We consider the Cauchy problem of the modified KdV (mKdV) equation. Local wellposedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q} (\mathbb{R}) (2 \leq q \leq \infty)$. Moreover, we show that the data-to-solution map fails to be $C^3$ continuous in $M^s_{2,q} (\mathbb{R})$ when $s \lt 1/4$. We notice that $H^{1/4}$ is the critical Sobolev space for mKdV such that it is well-posed

An asymptotic analysis of wave propagation in randomly perturbed waveguides is carried out in order to identify the effective Markovian dynamics of the guided mode powers. The main result consists in a quantification of the fluctuations of the mode powers and wave intensities that increase exponentially with the propagation distance. The exponential growth rate is studied in detail so as to determine

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

Solutions of first-order nonlinear hyperbolic conservation laws typically develop shocks in finite time even from smooth initial conditions. However, in heterogeneous media with rapid spatial variation, shock formation may be delayed or avoided. When shocks do form in such media, their speed of propagation depends on the material structure. We investigate conditions for shock formation and propagation

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

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. We prove the existence and uniqueness of global classical solutions for $\rho_0 \in C^{3,\alpha} (I)$. Moreover, we also obtain the global existence of weak solutions and unique strong solutions for

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 inviscid Burgers’ equation. The approach is to ingeniously add artificial viscosity to construct approximate solutions

In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski’s coagulation-diffusion equations with nonhomogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer’s disease. In contrast

We establish the vanishing viscosity limit of viscous Burgers–Vlasov equations for one dimensional kinetic model about interactions between a viscous fluid and dispersed particles by using compensated compactness technique and the evolution of level sets arguments. The limit we obtained is exactly a finite-energy weak solution to the inviscid equations.

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

A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that such cell population model converges at the incompressible limit towards a Hele-Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint. This constraint is important to be satisfied in many

We construct a mean-field variational model to study how the dependence of dielectric coefficient (i.e., relative permittivity) on local ionic concentrations affects the electrostatic interaction in an ionic solution near a charged surface. The electrostatic free-energy functional of ionic concentrations, which is the key object in our model, consists mainly of the electrostatic potential energy and