In this paper, we consider the following magnetic nonlinear Choquard equation $$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda f(u)\ \text { in }\ \mathbb {R}^N, \end{aligned}$$ where \(2_{\alpha }^{*}=\frac{2N-\alpha }{N-2}\) is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, \(\lambda

This paper reports the possibilities of predicting buckling of thin shells with nondestructive techniques during operation. It examines shallow shells fabricated from high-strength materials. Such structures are known to exhibit surface displacements exceeding the thickness of the elements. In the explored shells, relaxation oscillations of significant amplitude could be generated even under relatively

In this article, the authors investigate the non-autonomous magneto-micropolar fluids in a two-dimensional bounded domain. They first prove the existence of a pullback attractor for the associated process. Then, they construct a family of invariant Borel probability measures supported on the pullback attractor and prove that this family of probability measures is indeed a statistical solution for the

The Poisson–Boltzmann (P–B) equation is of fundamental importance in understanding solid–liquid electrolyte interfaces that are present in many fields. Due to the nonlinearity, it is usually challenging to find the explicit exact solutions of the P–B equation. The present work reports several perturbation solutions for the nonlinear P–B equation in the Cartesian and spherical coordinates. The new solutions

In this article, we study the stabilization problem for a hyperbolic-type Stokes system posed on a bounded domain. We show that when the damping effects are restricted to a subdomain satisfying the geometric control condition, the energy of the system decays exponentially. The result is a consequence of a new quasi-mode estimate for the Stokes system.

Stationary incompressible Newtonian fluid flow governed by external force and external pressure is considered in a thin rough pipe. The transversal size of the pipe is assumed to be of the order \(\varepsilon \), i.e., cross-sectional area is about \(\varepsilon ^{2}\), and the wavelength in longitudinal direction is modeled by a small parameter \(\mu \). Under general assumption \(\varepsilon ,\mu

This paper aims to clarify the homogenization results of the fluid–structure interaction in porous structures under the quasi-static and dynamic loading regimes. In the latter case, the acoustic fluctuations yield naturally a linear model which can be introduced in the configuration deformed as the consequence of the steady permanent flow. We consider a Newtonian slightly compressible fluid under the

We prove the existence and uniqueness of global strong solutions to the Maxwell–Navier–Stokes system in a two dimensional bounded domain with Navier-type boundary condition for the velocity.

Various non-local structural derivative diffusion models have been proposed based on different kernel functions to describe the anomalous time dependence of the mean-squared displacements. In the present study, the fundamental solutions for constant and variable-order structural derivative advection–dispersion models are achieved via scaling transformation and the generalized non-Euclidean Hausdorff

In this paper, we establish the global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in critical space by constructing the current function \(J=\nabla \times B\) as an additional unknown. Meanwhile, we consider the long-time behavior of the solutions and get some decay estimates.

This paper considers a one-dimensional generalized Allen–Cahn equation of the form $$\begin{aligned} u_t = \varepsilon ^2 (D(u)u_x)_x - f(u), \end{aligned}$$ where \(\varepsilon > 0\) is constant, \(D = D(u)\) is a positive, uniformly bounded below, diffusivity coefficient that depends on the phase field u, and f(u) is a reaction function that can be derived from a double-well potential with minima

This paper deals with the solution of two-species chemotaxis system $$\begin{aligned} \left\{ \begin{array}{llll} u_t=d_1\Delta u-\chi _1\nabla \cdot (u\nabla w)+\mu _{1}u(1-u-a_{1}v),&{}\quad x\in \Omega ,\quad t>0,\\ v_t=d_2\Delta v-\chi _2\nabla \cdot (v\nabla w)+\mu _{2}v(1-a_{2}u-v),&{}\quad x\in \Omega ,\quad t>0,\\ 0=d_3\Delta w-\gamma w+\alpha u+\beta v&{}\quad x\in \Omega ,\quad t>0 \end{array}

The evolution of a cloud of particles in a compressible fluid can be modeled with a Vlasov–Fokker–Planck equation for the distribution function of the particles coupled with Navier–Stokes or Euler equations for the density and velocity of the fluid. Formal calculations have established the convergence of solution to the mesoscopic model to solutions to the macroscopic Navier–Stokes or Euler model coupled

In this paper we consider the following quasilinear Schrödinger–Poisson system $$\begin{aligned} \left\{ \begin{array}[c]{ll} - \Delta u +u+\phi u = \lambda f(x,u)+|u|^{4}u &{}\ \text{ in } {\mathbb {R}}^{3} \\ -\Delta \phi -\varepsilon ^{4} \Delta _4 \phi = u^{2} &{} \ \text{ in } {\mathbb {R}}^{3}, \end{array} \right. \end{aligned}$$ depending on the two parameters \(\lambda ,\varepsilon >0\). We

This work studies the existence of positive solutions for fractional Schrödinger equations of the form $$\begin{aligned} (-\Delta )^s u + V(x) u = g(u)+\lambda |u|^{q-2}u\quad \text{ in }\quad {\mathbb {R}}^N, \end{aligned}$$ where \(s\in (0,1)\), \(N>2s\), \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a potential function which can vanish at infinity, \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\)

This work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by a nonlinear splitting of tensile and compressive strains, featuring non-interpenetration of the fracture lips. The alternating scheme is coupled with an \(L^{2}\)-penalization in the phase field variable

In this paper, we study the initial boundary value problem for the system \(\Delta v= u_{x_1},\ u_t-\text{ div }\left( \left( (a|{\mathbf {q}}|+m)I+(b-a)\frac{{\mathbf {q}}\otimes {\mathbf {q}}}{|{\mathbf {q}}|}\right) \nabla u\right) =-\nabla u\cdot {\mathbf {q}}\), where \({\mathbf {q}}=(-v_{x_2}, v_{x_1})^{\mathrm{T}}, {\mathbf {q}}\otimes {\mathbf {q}}={\mathbf {q}}{\mathbf {q}}^{\mathrm{T}}\)

The current paper is devoted to the stochastic tamed 3D Naiver-Stokes equation with degenerate noise with degenerate random forcing. We prove that the associated Markov semigroup possesses an exponentially attracting invariant measure by showing that the corresponding dynamical system possesses a strong type of Lyapunov structure along with a special type of gradient inequality and is of a relatively

In this communication a spectral constitutive equation for nonlinear viscoelastic-electroactive bodies with short-term memory response is developed, using the total stress formulation and the electric field as the electric independent variable. Spectral invariants, each one with a clear physical meaning and hence attractive for use in experiment, are used in the constitutive equation. A specific form

This paper deals with a model for solids with porous channels filled by an incompressible isotropic fluid. The Darcy–Brinkman–Stokes law is derived, that represents a rate equation for the local mass flux of the fluid, presenting relaxation times in which this flux evolves towards its local-equilibrium value, viscous effects and a permeability tensor referring to a response of the system to an external

Mathematical models that describe the tumor growth process have been formulated by several authors in order to understand how cancer develops and to develop new treatment approaches. In this study, it is aimed to investigate the long-time behavior of the two-phase diffuse-interface model, which was proposed in Hawkins-Daarud et al. (Int J Numer Methods Biomed Eng 28:3–24, 2012) to model a tumor tissue

This paper is devoted to analytical solutions for the base flow and temporal stability of a liquid film driven by gravity over an inclined plane when the fluid rheology is given by the Carreau–Yasuda model, a general description that applies to different types of fluids. In order to obtain the base state and critical conditions for the onset of instabilities, two sets of asymptotic expansions are proposed

We are concerned with the vanishing viscosity limit of the 2D compressible micropolar equations to the Riemann solution of the 2D Euler equations which admit a planar rarefaction wave. In this article, the key point of the analysis is to introduce the hyperbolic wave, which helps us obtain the desired uniform estimates with respect to the viscosities. Moreover, the proper combining of rotation terms

We consider an improved Nernst–Planck–Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non-equilibrium. The elastic deformation of the medium, that induces an inherent coupling of mass and momentum transport, is taken into account. The model consists of convection–diffusion–reaction equations for the constituents of the mixture, of the Navier–Stokes

In this paper, we introduce a new Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, governed by an approximating function h. We characterize the well-posedness in terms of the metric properties of the family of approximating sets, under various assumptions on h. Then, we use the well-posedness properties in order to obtain convergence results of the solution with respect

We study the structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical

We consider positive solutions to the weighted elliptic problem $$\begin{aligned} -\text{ div } (|x|^\theta \nabla u)=|x|^\ell u^p \;\;\text{ in } {\mathbb {R}}^N \backslash {{\overline{B}}},\quad u=0 \;\; \text{ on } \partial B, \end{aligned}$$ where B is the standard unit ball of \({\mathbb {R}}^N\). We give a complete answer for the existence question for \(N':=N+\theta >2\) and \(p > 0\). In particular

This paper is devoted to the chemotaxis model with indirect production and general kinetic function $$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u),\qquad&x\in \Omega ,\,t>0,\\&v_t=\Delta v-v+w,\qquad&x\in \Omega ,\,t>0,\\&\tau w_t+\lambda w=g(u),\qquad&x\in \Omega ,\,t>0, \end{aligned} \right. \end{aligned}$$ in a bounded domain \(\Omega \subset \mathbb {R}^n(n\le

The problem of constructing an exact solution of singular integro-differential equations related to problems of adhesive interaction between elastic thin semi-infinite homogeneous patch and elastic plate is investigated. For the patch loaded with horizontal forces the usual model of the uniaxial stress state is valid. Using the methods of the theory of analytic functions and integral transformation

The formulation of the nonlinear problem corresponding to the process of stationary heat conduction in homogeneous triaxial ellipsoid with increasing temperature and intensity of volumetric energy release was used to build a variational form of a mathematical model of this process. This form includes a functional defined on a set of continuous and piecewise differentiable functions that approximate

The interfacial Stoneley waves at the Wiechert condition imposed on physical properties of the adjacent isotropic and homogeneous halfspaces are analyzed. It is shown both theoretically and numerically that the speed of propagation of Stoneley waves along the Wiechert line is a monotonically decreasing function of the dimensionless Wiechert parameter. It is also found that such a monotonic behavior

We consider a standard Navier–Stokes system on the n-D torus \({\mathbb {T}}^n={\mathbb {R}}^n/{\mathbb {Z}}^n\), valid when the flow is not very compressible and the temperature does not vary too much. We construct a sequence of approximate solutions that tend to satisfy the equations in a weak sense for arbitrary physical initial conditions. By weak compactness, we obtain Radon measure solutions

In this paper, we aim at a reduced 2d-model describing the observable states of the magnetization in curved thin films. Under some technical assumptions on the geometry of the thin-film, it is well-known that the demagnetizing field behaves like the projection of the magnetization on the normal to the thin film. We remove these assumptions and show that the result holds for a broader class of surfaces;

This paper studies a Riemann problem for the isentropic Euler equations and addresses two gaps in the literature concerning positive pressures. Such study is made using a product of distributions and a solution concept that extends the classical solution concept. Under certain conditions, even for positive pressures, it is shown that the Riemann problem has solutions, which are \(\delta \)-shock waves

We give a series of very general sufficient conditions in order to ensure the uniqueness of positive large solutions for \(-\Delta u+f(x,u)=0\) in a bounded domain \(\Omega \) where \(f:{{\overline{\Omega }}}\times {\mathbb {R}}\mapsto {\mathbb {R}}_+\) is a continuous function, such that \(f(x,0)=0\) for \(x\in {{\overline{\Omega }}}\), and \(f(x,r)>0\) for x in a neighborhood of \(\partial \Omega

This paper concerns the lower bound decay rate of global solution for compressible Navier–Stokes–Korteweg system in three-dimensional whole space under the \(H^{4}\times H^{3}\) framework. At first, the lower bound of decay rate for the global solution converging to constant equilibrium state (1, 0) in \(L^2\)-norm is \((1+t)^{-\frac{3}{4}}\) if the initial data satisfy some low-frequency assumption

This paper concerns the Cauchy problem of the three-dimensional nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with density-dependent viscosity and vacuum. We first establish some key a priori algebraic decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence and large time asymptotic behavior of strong solutions in the whole

We seek solutions for non-cooperative elliptic systems of two Schrödinger equations which can model phenomena in Physics and Biology. General conditions are assumed under the potentials, which produce convenient spectral properties on the elliptic operator concerned; hence, the non-cooperation characterizes it as a strongly indefinite problem. Spectral theory is employed along with an abstract linking

A hydrodynamic model, with incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid, with the upper layer exposed to a free surface. Following the Euler–Bernoulli beam equation, the elastic bed is approximated as a thin elastic plate. The surface tension at the interface of the layers is completely ignored

In this paper, the time-periodic solution to the compressible viscous quantum magnetohydrodynamic model in a periodic domain is studied. Under the boundedness assumption on the external force, we prove the existence of the time-periodic solution by using the topological degree theory and parabolic regularization method. Furthermore, the uniqueness of the time-periodic solution is shown.

The Biot–Savart law is used in aerodynamic theory to calculate the velocity induced by curved vortex lines. Explicit formulas are developed, using multivariate Appell hypergeometric functions, for the velocity induced by a general parabolic vortex segment. The formulas are derived by constructing a particular pencil of elliptic curves whose period integrals provide the solution to the induced velocity

This paper aims at providing a small-volume expansion framework for the scattering resonances of an open cavity perturbed by small particles. The shift of the scattering resonances induced by the small particles is derived without neglecting the radiation effect. The formula holds for arbitrary-shaped particles. It shows a strong enhancement in the scattering resonance shift in the case of subwavelength

Viscous flow of fluid film on outer surface of a sloped rotating cylinder in gravitational field is studied. Thickness of the fluid layer is assumed to be small compared to the cylinder radius, which allows asymptotic analysis. Governing equation for the thickness dynamics is derived. The equation accounts for viscous effects, gravity, centrifugal and capillary forces. A criterion for existence of

An investigation on the dispersion of a solute in the flow of blood through a constricted artery with an absorptive wall having relevance to arterial pharmacokinetics is carried out in which the rheology of blood is characterized by the Casson model. The solute is considered to be administered at the inlet of the finite arterial lumen. The governing equations of motion for the flow and the convection–diffusion

The yield behavior of crystalline solids is determined by the motion of defects like dislocations, twin boundaries and coherent phase boundaries. These solids are hardened by introducing precipitates—small particles of a second phase. It is generally observed that the motion of line defects like dislocations are strongly inhibited or pinned by precipitates while the motion of surface defects like twin

This paper deals with a two-dimensional quasilinear chemotaxis-growth system with indirect signal consumption $$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla v)+\mu (u-u^{2}),&(x,t)\in \Omega \times (0,\infty ), \\&v_t=\Delta v-vw,&(x,t)\in \Omega \times (0,\infty ), \\&w_t=-\delta w+u,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

In this paper, we consider the tropical climate model with temperature-dependent viscosity in \({\mathbb {R}}^{2}\). We obtain the global strong solution provided that the initial data \(\Vert v_{0}\Vert _{H^{s}}^{2}+\Vert \theta _{0}\Vert _{H^{s}}^{2}(s>1)\) are small enough. It is worth to point out that there is no any smallness condition on the barotropic mode u.

Consideration in the present paper is a weakly dissipative shallow water equation. The parameters take different values, which include several different important shallow water equations, such as CH equation, DP equation, Novikov equation and so on. The wave-breaking phenomena are investigated by three different kinds of method. Due to the presence of high-order nonlinear terms \(u^{2n+1}\) and \(u^{2m}u_{xx}\)

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution (u, b) becomes regular provided that one velocity and one current density component of the solution satisfy $$\begin{aligned} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( {\mathbb {R}}^{3}\right) \right) \quad \text { with }\frac{45}{7}

We introduce a one-dimensional stress-rate type nonlinear viscoelastic model for solids that obey the assumptions of the strain-limiting theory. Unlike the classical viscoelasticity theory, the critical hypothesis in the present strain-limiting theory is that the linearized strain depends nonlinearly on the stress and the stress rate. We show the thermodynamic consistency of the model using the complementary

The final purpose of this paper is to show that, by inserting a convexity constraint on the cables of a suspension bridge, the torsional instability of the deck appears at lower energy thresholds. Since this constraint is suggested by the behavior of real cables, this model appears more reliable than the classical ones. Moreover, it has the advantage to reduce to two the number of degrees of freedom

We explore the efficiency of a thermoelectric energy converter constituted by a Si/Ge nanowire of length L. A constitutive equation of thermal conductivity as function of composition and temperature is derived in accordance with experimental data obtained at the constant temperatures \(T=300\,\hbox {K}\), \(T=400\,\hbox {K}\), and \(T=500\,\hbox {K}\) by a nonlinear regression method. A thermodynamic

We consider the large time behavior of weak solutions to the 3D magneto-micropolar fluid system. We establish the rate of decay \(\bigl \Vert \;\!({{\varvec{u}}},{{\varvec{w}}}, {{\varvec{b}}})(\cdot ,t)\,\!\bigr \Vert _{{{\varvec{L}}}^{2}({\mathbb {R}}^3)} \,\le \,C \;\!(t + 1)^{-\;\!\frac{3}{4}}\) for all \(t \ge 0\) through Fourier splitting method. Furthermore, we use a direct method to show how

We are concerned with the Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \rho _t+u\cdot \nabla \rho =\Delta \rho -\nabla \cdot (\rho \mathcal {S}(x,\rho ,c)\nabla c)-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ m_t+u\cdot \nabla m=\Delta m-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ c_t+u\cdot \nabla c=\Delta c-c+m, &{}\quad (x,t)\in \varOmega

In this paper, we establish a small data global well-posedness results for 3D incompressible MHD equations with Hall and ion-slip effects. Then, by using Fourier splitting method and iteration, we consider the decay rate on higher order derivative of solutions.

We establish a large deviation principle for the 2D stochastic Cahn–Hilliard–Navier–Stokes equations driven by multiplicative noise in a bounded domain. This system consists of the Navier–Stokes equations for the velocity, coupled with a Cahn–Hilliard model for the order parameter. The proof is completed using a weak convergence approach based on the variational representational of functional of infinite-dimensional

Conjugation of reciprocal transformations is used to solve a class of boundary value problems involving a source term relevant to water transport through a heterogeneous medium with a volumetric extraction mechanism. The main emphasis is the solution method that involves conjugation of reciprocal transformations, as well as other changes of variable, applied to a newly identified integrable model.

In this paper, we formulate a theory for the coupling of accretion mechanics and thermoelasticity. We present an analytical formulation of the thermoelastic accretion of an infinite cylinder and of a two-dimensional block. We develop numerical schemes for the solution of these two problems, and numerically calculate residual stresses and observe a strong dependence of the final mechanical state on

This paper is concerned with a barotropic model of capillary compressible fluids derived by Dunn and Serrin (1985). We consider the initial boundary value problem in bounded domains of \({\mathbb {R}}^d (d=2,3)\) with more general pressure including Van der Waals equation of state. First, the local in time existence and uniqueness of strong solutions is proved for initial data belonging to the Sobolev

It is known that in the case that several constitutive tensors fail to be positive definite the system of the thermoelasticity could become unstable and, in certain cases, ill-posed in the sense of Hadamard. In this paper, we consider the Moore–Gibson–Thompson thermoelasticity in the case that some of the constitutive tensors fail to be positive and we will prove basic results concerning uniqueness