Narrow Results

The Klein–Gordon equation with quadratic nonlinearity is frequently used to model various problems of physics. On the other hand, it was shown that in many situations, the use of fractional derivatives instead of ordinary derivatives provides more realistic models. In this work, a one-dimensional sequential time-fractional Klein–Gordon equation posed on a bounded interval, subject to certain initial and boundary conditions, is investigated. The fractional derivatives are considered in the Caputo sense and a weight function $h>0$ is allowed in front of the quadratic nonlinearity. Namely, we first establish general sufficient conditions for the nonexistence of solutions. Next, some particular cases of weight functions $h$ are discussed. In our approach, we make use of the test function method. This method requires an appropriate choice of a family of test functions. In this paper, we make a judicious choice of test functions taking into consideration the nonlocal property of the Caputo fractional derivative, the initial and boundary conditions, and the geometry of the domain.

In this paper, we consider a periodic Hunter–Saxton equation with weak dissipation. We obtain the existence of global weak solutions to the equation for any initial data in H¹(𝕊).

This paper is concerned with the rotation-two-component Camassa–Holm (R2CH) system, which is a model for equatorial water waves under the influence of the Coriolis force. The system includes the Dullin–Gottwald–Holm equation, the standard two-component integrable Camassa–Holm (CH) system and the CH equation as special cases. We aim to explore whether the R2CH system admits some generalized peakon weak solutions in the sense of distribution. The exact weak solutions in a particular form are derived by the ansatz method and they are proven to be the generalized peakon weak solutions. These solutions may help to explain the wave-breaking phenomenon in the wave motion to some extent. The method proposed in this paper is effective, which might be applied to study other nonlinear wave equations directly and even to be extend to search for fractal solitary wave.

Under investigation in this paper is a two-component Novikov system (also called Geng-Xue equation), which was proposed by Geng and Xue in 2009. Firstly, via the Lie symmetry method, infinitesimal generators, commutator table of Lie algebra and symmetry groups of the two-component Novikov system are presented. At the same time, some group invariant solutions are computed through similarity reductions. In particular, we construct peakon solution by applying the distribution theory. In addition, based on obtained group invariant solutions and symmetry transformations, we derive some new exact solutions, which include stationary solutions, smooth solutions, and a weak solution. The analytical properties to some of group invariant solutions and new exact solutions are discussed, such as decay, asymptotic behavior, and boundedness.

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.

We study a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is static, the material behavior is described with a linearly electro-elastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is in the form of a system of two coupled hemivariational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on an abstract result on operator inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our results are valid.

We consider a reaction–diffusion system of 2 × 2 equations modeling the spread of early tumor cells. The existence of weak solutions is ensured by a classical argument of Faedo–Galerkin method. Then, we present a numerical scheme for this model based on a finite volume method. We establish the existence of discrete solutions to this scheme, and we show that it converges to a weak solution. Finally, some numerical simulations are reported with pattern formation examples.

Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities describing evolution of magnetization, temperature, magnetic field, and electric field in electrically-conductive unsaturated ferromagnets. The system is derived from a recently-proposed thermodynamically-consistent continuum theory for the ferro/paramagnetic transition. Besides the standard viscous-like damping, dissipation due to eddy currents and domain-wall pinning is considered.

We consider a problem modeling the steady flow of a compressible heat conducting Newtonian fluid subject to the slip boundary condition for the velocity. Assuming the pressure law of the form p(ϱ, ϑ) ~ ϱ^γ + ϱϑ, we show (under additional assumptions on the heat conductivity and the viscosity) that for any γ > 1 there exists a variational entropy solution to our problem (i.e. the weak formulation of the total energy balance is replaced by the entropy inequality and the global total energy balance). Moreover, if (together with further restrictions on the heat conductivity), the solution is in fact a weak one. The results are obtained without any restriction on the size of the data.

The paper is concerned with a class of mathematical models for polymeric fluids, which involves the coupling of the Navier–Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic–hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient, appearing in the balance of linear momentum equation in the Navier–Stokes system, includes dependence on the shear rate as well as on the weight-averaged polymer chain length. The system of partial differential equations under consideration captures the impact of polymerization and depolymerization effects on the viscosity of the fluid. We prove the existence of global-in-time, large-data weak solutions under fairly general hypotheses.

A compressible FENE-P-type model with stress diffusion is derived from an approximate macroscopic closure of a compressible Navier–Stokes–Fokker–Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent are idealised as pairs of massless beads connected with finitely extensible nonlinear elastic (FENE) springs. We develop a priori bounds for the model, including logarithmic bounds, which guarantee the non-negativity of the conformation tensor and a bound on its trace, and we prove the existence of large-data global-in-time finite-energy weak solutions in two and three space dimensions.

We are concerned with a system of partial differential equations (PDEs) describing internal flows of homogeneous incompressible fluids of Bingham type in which the value of activation (the so-called yield) stress depends on the internal pore pressure governed by an advection–diffusion equation. After providing the physical background of the considered model, paying attention to the assumptions involved in its derivation, we focus on the PDE analysis of the initial and boundary value problems. We give several equivalent descriptions for the considered class of fluids of Bingham type. In particular, we exploit the possibility to write such a response as an implicit tensorial constitutive equation, involving the pore pressure, the deviatoric part of the Cauchy stress and the velocity gradient. Interestingly, this tensorial response can be characterized by two scalar constraints. We employ a similar approach to treat stick-slip boundary conditions. Within such a setting we prove long-time and large-data existence of weak solutions to the evolutionary problem in three dimensions.

The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.

We prove that there exists a large-data and global-in-time weak solution to a system of partial differential equations describing the unsteady flow of an incompressible heat-conducting rate-type viscoelastic stress-diffusive fluid filling up a mechanically and thermally isolated container of any dimension. To overcome the principal difficulties connected with ill-posedness of the diffusive Oldroyd-B model in three dimensions, we assume that the fluid admits a strengthened dissipation mechanism, at least for excessive elastic deformations. All the relevant material coefficients are allowed to depend continuously on the temperature, whose evolution is captured by a thermodynamically consistent equation. In fact, the studied model is derived from scratch using only the balance equations for linear momentum and energy, the formulation of the second law of thermodynamics and the constitutive equation for the internal energy. The latter is assumed to be a linear function of temperature, which simplifies the model. The concept of our weak solution incorporates both the temperature and entropy inequalities, and also the local balance of total energy provided that the pressure function exists.

This paper mainly deals with a self-consistent chemotaxis-Navier–Stokes system with porous medium diffusion in a three-dimensional (3D) bounded and smooth domain. The novelty here is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered, which leads to stronger coupling than the usual chemotaxis-fluid model studied in most existing literatures. It is proved that for any suitably regular initial data, the associated no-flux/no-flux/Dirichlet problem possesses at least one global weak solution or global very weak solution. To the best of our knowledge, this is the first result on the global solvability of the 3D self-consistent chemotaxis-Navier–Stokes system with porous medium diffusion. Our results inter alia provide a more in-depth understanding on the chemotaxis-Navier–Stokes system, and significantly improve previously known ones.

In this paper, we study the following boundary value problem involving the weak $p$-Laplacian.

We establish Hölder continuity of weak solutions for degenerate quasilinear subelliptic systems with different weights, with weak Harnack inequalities obtained here and a generalization of Caffarelli's idea.

We consider a mathematical model which describes the frictional contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law in which the friction bound depends both on the tangential displacement and on the value of the penetration. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solutions of regularized problems is obtained. Next, we prove the convergence of the weak solutions of regularized problems to the weak solution of the initial nonregularized problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems. The solution of the resulting nonsmooth and nonconvex frictional contact problems is found, basing on approximation by a sequence of nonsmooth convex programming problems. Some numerical simulation results are presented in the study of an academic two-dimensional example.

We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property.

We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.

We consider general second-order linear elliptic partial differential equations having random coefficients and random data and fulfilling the homogeneous Dirichlet condition. We prove the existence and uniqueness of the weak solution in a certain tensor product space which is suitably completed to make it a Hilbert space. The factors of this space are a Sobolev space of functions depending on the space variable and a general Sobolev space of functions depending on the stochastic variable.