Narrow Results

In this work, we investigate the question of existence of weak solutions to the density dependent stochastic Navier–Stokes equations. The noise considered contains functions which depend nonlinearly on the velocity and which do not satisfy the Lipschitz condition. Furthermore, the initial density is allowed to vanish. We introduce a suitable notion of probabilistic weak solution for the problem and prove its existence.

We prove the existence and uniqueness of strong solutions of a three-dimensional system of globally modified Navier–Stokes equations with delay in the locally Lipschitz case. The asymptotic behavior of solutions, and the existence of pullback attractor are also analyzed.

In this work we propose a new numerical method for solving a thermomechanical coupled problem which arises as a mathematical model for the evolution of ice sheets profiles and the corresponding temperatures. From the accumulation–ablation ratio, the atmospheric temperature and the geothermic flux, the ice sheet profile is the solution of a moving boundary problem governed by a nonlinear convection–diffusion equation. Moreover, the mathematical model which governs the temperature distribution involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two-phase Stefan formulation of the polythermal regime. The problems are discretized in time with upwind characteristics schemes and in space with finite elements. The nonlinearities are solved either by fixed point methods or by duality techniques. Several numerical simulation examples involving real data sets issued from the Antartic ice sheet are shown.

We study an initial-boundary value problem in an infinite viscoelastic strip with a semi-infinite fixed crack. For this problem we prove the existence and uniqueness of a weak solution which is prescribed on each side of the extended crack in Sobolev-type spaces.

The equations for steady flows of Herschel–Bulkley fluids are considered and the existence of a weak solution is proved for the Dirichlet boundary-value problem. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous constitutive relation between the Cauchy stress and the symmetric part of the velocity gradient. Such a fluid stiffens if its local stresses do not exceed τ*, and it behaves like a non-Newtonian fluid otherwise. We address here a class of nonlinear fluids which includes shear-thinning p-law fluids with 9/5 < p ≤ 2. The flow equations are formulated in the stress-velocity setting (cf. Ref. 25). Our approach is different from that of Duvaut–Lions (cf. Ref. 10) developed for classical Bingham visco-plastic materials. We do not apply the variational inequality but make use of an approximation of the Herschel–Bulkley fluid with a generalized Newtonian fluid with a continuous constitutive law.

In this paper we propose a new space and time finite elements discretization of the Landau–Lifshitz equations which may be readily used for numerical computations. We then prove its convergence to a weak solution in the sense given by Alouges and Soyeur or Labbé in the literature.

Recently, Berthelin et al.,⁴ introduced a traffic-flow model describing the formation and the dynamics of traffic jams. This model which consists of a Constrained Pressureless Gas Dynamics system assumes that the maximal density constraint is independent of the velocity. However, in practice, the distribution of vehicles on a highway depends on their velocity. In this paper, a more realistic model namely the Second Order Model with Constraints (in short SOMC) is proposed, derived from the Aw and Rascle model,¹ which takes into account this feature. Moreover, when the maximal density constraint is saturated, the SOMC model "relaxes" to the Lighthill and Whitham model.²⁰ An existence result of weak solutions for this model by means of cluster dynamics is proved in order to construct a sequence of approximations, and the associated Riemann problem is solved completely.

The aim of this paper is to study the damage evolution in an elasto-piezoelectric body. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, and the piezoelectric effects are included into the model. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is then proved by using the theory of maximal monotone operators, the Schauder fixed-point theorem, and a comparison result.

We consider the steady compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain. We prove the existence of a solution for arbitrarily large data under the assumption that the pressure p(ϱ, θ) ~ ϱθ + ϱ^γ for assuming either the slip or no-slip boundary condition for the velocity and the Newton boundary condition for the temperature. The regularity of solutions is determined by the basic energy estimates, constructed for the system.

This paper is concerned with the mathematical analysis of a coupled elliptic–parabolic system modeling the interaction between the propagation of electric potential and subsequent deformation of the cardiac tissue. The problem consists in a reaction–diffusion system governing the dynamics of ionic quantities, intra- and extra-cellular potentials, and the linearized elasticity equations are adopted to describe the motion of an incompressible material. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we prove existence of weak solutions to the underlying coupled reaction–diffusion system and uniqueness of regular solutions. The proof of existence is based on a combination of parabolic regularization, the Faedo–Galerkin method, and the monotonicity-compactness method of Lions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with a numerical example illustrating the convergence of the method and some features of the model.

This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen–Leslie system of liquid crystal models in ${ℝ}^2$, with both general Leslie stress tensors and general Oseen–Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant. One of the main ideas of our proof is to perform suitable energy estimates at the level one order lower than the natural basic energy estimates for the Ericksen–Leslie system.

Radon-measure-valued solutions to a size structured population model of the McKendrick–von Foerster-type are analytically studied under general assumptions on individuals’ growth, birth and mortality rates. The model is used to describe changes in size structure of zooplankton when prey size-dependent mortality rate is a consequence of a planktivorous fish foraging in low prey-density environment (commonly found in predator-controlled populations). The model of foraging is based on the optimization of the rate of net energy intake as a function of predator speed. Mortality is defined as an operator on a metric space of nonnegative Radon measures equipped with the bounded Lipschitz distance. The solutions to the size structured model of zooplankton population are studied analytically and numerically. Numerical solutions (derived using the Escalator Boxcar Train (EBT)-like schema), in particular those starting from Dirac deltas corresponding to distinct cohorts, exhibit regularization in time and convergence to the same stationary state.

A two-dimensional mathematical model for cells migrating without adhesion capabilities is presented and analyzed. Cells are represented by their cortex, which is modeled as an elastic curve, subject to an internal pressure force. Net polymerization or depolymerization in the cortex is modeled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility.

A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent.

We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone $p$-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.

We consider an alternative Navier–Stokes model for compressible viscous ideal gases, originally proposed in [M. Svärd, A new Eulerian model for viscous and heat conducting compressible flows, Physica A 506 (2018) 350–375.] We derive a priori estimates that are sufficiently strong to support a weak entropy solution of the system. Guided by these estimates, we propose a finite-volume scheme, derive the analogous estimates and demonstrate grid convergence towards a weak entropy solution. Furthermore, this existence proof is valid for “large” initial data and no a priori assumptions on the solution are needed.

The thermodynamic model of viscoelastic deformable magnetic materials at finite strains is formulated in a fully Eulerian way in rates. The Landau theory applied to a ferro-to-paramagnetic phase transition, the gradient theory (due to an exchange energy) for magnetization with general mechanically dependent coefficient, hysteresis in magnetization evolution by the Gilbert equation involving an objective corotational time derivative of magnetization, and the demagnetizing field are considered in the model. The Kelvin–Voigt viscoelastic rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. An existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization.

We study a nonlocal Neumann problem driven by a nonhomogeneous elliptic differential operator. The reaction term is a nonlinearity function that exhibits p-superlinear growth but need not satisfy the Ambrosetti–Rabinowitz condition. By using an abstract linking theorem for smooth functionals, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example illustrates the main abstract result of this paper.

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions

In this paper, we study a Schrödinger–Bopp–Podolsky (SBP) system of partial differential equations in a bounded and smooth domain of ${ℝ}^3$ with a nonconstant coupling factor. Under a compatibility condition on the boundary data we deduce existence of solutions by means of the Ljusternik–Schnirelmann theory.