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 [O.A. Oleinik, Dokl. Akad. Nauk SSSR, 150(4):28–31, 1963] on the tangential velocity

In this paper, we establish a new criterion for the orbital stability of periodic waves related to a general class of regularized dispersive equations. More specifically, we present sufficient conditions for the stability without knowing the positiveness of the associated Hessian matrix. As application of our method, we show the orbital stability for the fifth-order model. The orbital stability of

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 a conditional law of a partially observed Markov diffusion process. It then follows that the adjoint time-inverse forward backward doubly

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

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 are prescribed in the form of 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

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 $f_\ast \varrho$ which we would like to study. However, we might not have access to $f$, but to its approximation $g$. This problem is common in the use of surrogate models for numerical

We study the Cauchy problem of density-dependent magnetic Bénard system with zero density at infinity on the whole two-dimensional (2D) space. Despite the degenerate nature of the problem, we show the local existence of a unique strong solution in weighted Sobolev spaces by energy method.

We consider a one-dimensional chain of coupled oscillators in contact at both ends with heat baths at different temperatures, and subject to an external force at one end. The Hamiltonian dynamics in the bulk is perturbed by random exchanges of the neighbouring momenta such that the energy is locally conserved. We prove that in the stationary state the energy and the volume stretch profiles, in large

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

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

In this paper, we investigate the smooth solutions for the antiferromagnets Landau–Lifshitz–Bloch (LLB) equation with periodic initial value, which can describe the dynamics of micromagnets under high temperature. The existence and uniqueness of smooth solutions for LLB equation in $\mathbb{R}^2$ and $\mathbb{R}^3$ are proved.

We consider the Cauchy problem to the 1D non-resistive compressible magnetohydrodynamics (MHD) equations. We establish 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

In this paper, two approaches for modeling three-component fluid flows using diffuse 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 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

A method is presented to solve two-phase problems involving a material quantity on an interface. The interface can be advected, stretched, and change topology, and material can be adsorbed to or desorbed from it. The method is based on the use of a diffuse interface framework, which allows a simple implementation using standard finite-difference or finite-element techniques. Here, finite-difference

We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated