Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 2013"

Sharp nonexistence results of prescribed L 2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations

Abstract: In this paper, we study the existence of minimizers for $$F(u) = \frac{1}{2} \int_{\mathbb{R}^3} | abla u|^{2} {\rm d}x + \frac{1}{4} \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{| u(x)|^2 | u(y)|^2}{| x-y|} {\rm d}x{\rm d}y-\frac{1}{p} \int_{\mathbb{R}^3}|u|^p {\rm d}x$$ on the constraint $$S(c) = \{u \in H^1(\mathbb{R}^3) : \int_{\mathbb{R}^3}|u|^2 {\rm d}x = c\}$$ , where c > 0 is a given parameter. In the range $${p \in [3,\frac{10}{3}]}$$ , we explicit a threshold value of c > 0 separating existence and nonexistence of minimizers. We also derive a nonexistence result of critical points of F(u) restricted to S(c) when c > 0 is sufficiently small. Finally, as a by-product of our approaches, we extend some results of Colin et al. (Nonlinearity 23(6):1353–1385, 2010) where a constrained minimization problem, associated with a quasi-linear equation, is considered.

Global existence and convergence rates of smooth solutions for the full compressible MHD equations

Abstract: In this paper, we consider the global smooth solutions and their decay for the full compressible magnetohydrodynamic equations in R 3. We prove the global existence of smooth solutions near the constant state in Sobolev norms by energy method and show the convergence rates of the L p -norm of these solutions to the constant state when the L q -norm of the perturbation is bounded.

Solutions of half-space and half-plane contact problems based on surface elasticity

Abstract: Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.

Electromagnetism on anisotropic fractal media

Abstract: Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green–Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampere laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.

A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces

Abstract: We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient-flow formulation for electro-reaction–diffusion systems with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.

Stability of an abstract system of coupled hyperbolic and parabolic equations

Abstract: In this paper, we provide a complete stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$\begin{array}{ll}\;\;u_{tt} = -Au + \gamma A^{\alpha} \theta,\\ \quad \theta_t = -\gamma A^{\alpha}u_t - kA^{\beta}\theta,\\ u(0) = u_0, \quad u_t(0) = v_0, \quad \theta(0) = \theta_0\end{array}$$ where A is a self-adjoint, positive definite operator on a Hilbert space H. For \({(\alpha,\beta) \in [0,1] \times [0,1]}\) , the region of exponential stability had been identified in Ammar-Khodja et al. (ESAIM Control Optim Calc Var 4:577–593,1999). Our contribution is to show that the rest of the region can be classified as region of polynomial stability and region of instability. Moreover, we obtain the optimality of the order of polynomial stability.

On entropy weak solutions of Hughes’ model for pedestrian motion

Abstract: We consider a generalized version of Hughes’ macroscopic model for crowd motion in the one-dimensional case. It consists in a scalar conservation law accounting for the conservation of the number of pedestrians, coupled with an eikonal equation giving the direction of the flux depending on pedestrian density. As a result of this non-trivial coupling, we have to deal with a conservation law with space–time discontinuous flux, whose discontinuity depends non-locally on the density itself. We propose a definition of entropy weak solution, which allows us to recover a maximum principle. Moreover, we study the structure of the solutions to Riemann-type problems, and we construct them explicitly for small times, depending on the choice of the running cost in the eikonal equation. In particular, aiming at the optimization of the evacuation time, we propose a strategy that is optimal in the case of high densities. All results are illustrated by numerical simulations.

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

Abstract: This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H1(Ω) × L2(∂Ω) with boundary data from L2(∂Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H1(Ω) into L2(Ω) or L2(∂Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.

Strain gradient solutions of half-space and half-plane contact problems

Abstract: General solutions for the problems of an elastic half-space and an elastic half-plane, respectively, subjected to a symmetrically distributed normal force of arbitrary profile are analytically derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. Mindlin’s potential function method and Fourier transforms are employed in the formulation, and the half-space and half-plane contact problems are solved in a unified manner. The specific solutions for the problems of a half-space/plane subjected to a concentrated normal force or a uniformly distributed normal force are obtained by directly applying the general solutions, which recover the existing classical elasticity-based solutions of the Flamant and Boussinesq problems as special cases. In addition, the indentation problems of an elastic half-space indented by a flat-ended cylindrical punch, a spherical punch, and a conical punch, respectively, are solved using the general solutions, leading to hardness formulas that are indentation size- and material microstructure-dependent. Numerical results reveal that the displacement and stress fields in a half-space/plane given by the current SSGET-based solutions are smoother than those predicted by the classical elasticity-based solutions and do not exhibit the discontinuity and/or singularity displayed by the latter. Also, the indentation hardness values based on the newly obtained half-space solution are found to increase with decreasing indentation radius and increasing material length scale parameter, thereby explaining the microstructure-dependent indentation size effect.

Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

Abstract: We consider a magnetic Schrodinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum.

Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary

Abstract: We study the compressible magnetohydrodynamic equations in a bounded smooth domain in \({{\mathbb{R}}^2}\) with perfectly conducting boundary, and prove the global existence and uniqueness of smooth solutions around a rest state. Moreover, the low Mach limit of the solutions is verified for all time, provided that the initial data are well prepared.

Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation

Abstract: In this paper, we consider a dynamical model of population biology which is of the classical Fisher type, but the competition interaction between individuals is nonlocal. The existence, uniqueness, and stability of the steady state solution of the nonlocal problem on a bounded interval with homogeneous Dirichlet boundary conditions are studied.

On a 3D isothermal model for nematic liquid crystals accounting for stretching terms

Abstract: In the present contribution, we study a PDE system describing the evolution of a nematic liquid crystals flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the velocity fieldu is ruled by the Navier–Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamics of the director fieldd is described by a variation of a parabolic Ginzburg–Landau equation with a suitable penalization of the physical constraint |d| = 1. Such equation accounts for both the kinematic transport by the flow field and the internal relaxation due to the elastic energy. The main aim of this contribution is to overcome the lack of a maximum principle for the director equation and prove (without any restriction on the data and on the physical constants of the problem) the existence of global in time weak solutions under physically meaningful boundary conditions on d and u.

Global well-posedness and the classical limit of the solution for the quantum Zakharov system

Abstract: In this paper, we study the quantum Zakharov system, which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The global well-posedness result of this system in the energy and above energy spaces is obtained in the case d = 1, 2, 3. Moreover, the classical limit behavior of the quantum Zakharov system is also investigated as the quantum parameter tends to zero.

Optimal rigid inclusion shapes in elastic bodies with cracks

Abstract: The paper concerns the control of rigid inclusion shapes in elastic bodies with cracks. Cracks are located on the boundary of rigid inclusions and in the bulk. Inequality type boundary conditions are imposed at the crack faces to guarantee mutual non-penetration. Inclusion shapes are considered as control functions. First we provide the problem formulation and analyze the shape sensitivity with respect to geometrical perturbations of the inclusion. Then, based on Griffith criterion, we introduce the cost functional, which measures the shape sensitivity of the problem with respect to the geometry of the inclusion, provided by the energy release rate. We prove existence of optimal shapes for the problem considered.

On the existence of dynamics in Wheeler–Feynman electromagnetism

Abstract: Wheeler–Feynman electrodynamics (WF) is an action-at-a-distance theory about world-lines of charges that in contrary to the textbook formulation of classical electrodynamics is free of ultraviolet singularities and is capable of explaining the irreversible nature of radiation. In WF, the world-lines of charges obey the so-called Fokker–Schwarzschild–Tetrode (FST) equations, a coupled set of nonlinear and neutral differential equations that involve time-like advanced as well as retarded arguments of unbounded delay. Using a reformulation of this theory in terms of Maxwell–Lorentz electrodynamics without self-interaction that we have introduced in a preceding work, we are able to establish the existence of conditional solutions. These conditional solutions solve the FST equations on any finite time interval with prescribed continuations outside of this interval. As a byproduct, we also prove existence and uniqueness of solutions to the Synge equations on the time half-line for a given history of charge world-lines.

One-dimensional nonlinear chromatography system and delta-shock waves

Abstract: The Riemann problem for the nonlinear chromatography system is considered. Existence and admissibility of δ-shock type solution in both variables are established for this system. By the interactions of δ-shock wave with elementary waves, the generalized Riemann problem for this system is presented, the global solutions are constructed, and the large time-asymptotic behavior of the solutions are analyzed. Moreover, by studying the limits of the solutions as perturbed parameter \({\varepsilon}\) tends to zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data.

An exact description of nonlinear wave interaction processes ruled by 2 × 2 hyperbolic systems

Abstract: By means of a reduction approach to 2 × 2 quasilinear hyperbolic homogeneous systems of first-order PDEs, a full and exhaustive analysis of nonlinear wave interactions is achieved. The alteration in the profile as well as in the wave time distortion of the emerging pulses caused by the interaction process is illustrated in detail through exact solutions of initial value problems. Canonical forms of 2 × 2 systems which allow for special (soliton-like) hyperbolic wave interactions and of interest in applications are also determined.

Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows

Abstract: In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (characteristic discontinuity), the Euler equations are of elliptic–hyperbolic composite-mixed type. In this paper, we further prove that such a transonic characteristic discontinuity solution is unique and L1–stable with respect to the small perturbation of the incoming supersonic flow in Lagrangian coordinates.

Existence of solutions for the 3D-micropolar fluid system with initial data in Besov-Morrey spaces

Abstract: In this paper, we show a local-in-time existence result for the 3D micropolar fluid system in the framework of Besov–Morrey spaces. The initial data class is larger than the previous ones and contains strongly singular functions and measures.

Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces

Abstract: This paper discusses a class of impulsive nonlocal stochastic functional integrodifferential inclusions in a real separable Hilbert space. The existence of mild solutions of these inclusions is determined under the mixed continuous and Caratheodory conditions by using Bohnenblust–Karlin’s fixed point theorem and fractional operators combined with approximation techniques. An example is provided to illustrate the theory.

Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

Abstract: The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a 2 ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.

Bifurcation of positive solutions of a nonlinear discrete fourth-order boundary value problem

Abstract: Let \({T \in \mathbb{N}}\) be an integer with T ≥ 4. We give a global description of the branches of positive solutions of the nonlinear boundary value problem of fourth-order difference equation of the form $$\begin{array}{lll}\Delta^4 u(t-2)&=&f(t,u(t),\Delta^2u(t-1)),\quad t\in \{2,\ldots, T\},\\u(0)=&u(T+2)=\Delta^2u(0)=\Delta^2u(T)=0,\end{array}$$ that is not necessarily linearizable. Our approach is based on Krein–Rutman theorem, topological degree theory, and global bifurcation techniques.

Stability of contact discontinuity for steady Euler system in infinite duct

Abstract: In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct \({\mathcal{N}_0}\) of infinite length in \({\mathbb{R}^2}\) with width W0 and consider two uniform subsonic flow \({{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}\) with different horizontal velocity in \({\mathcal{N}_0}\) divided by a flat contact discontinuity \({\Gamma_{cd}}\). And, we slightly perturb the boundary of \({\mathcal{N}_0}\) so that the width of the perturbed duct converges to \({W_0+\omega}\) for \({|\omega| 0 }\). Then, we prove that if the asymptotic state at left far field is given by \({{U_l}^{\pm}}\), and if the perturbation of boundary of \({\mathcal{N}_0}\) and \({\delta}\) is sufficiently small, then there exists unique asymptotic state \({{U_r}^{\pm}}\) with a flat contact discontinuity \({\Gamma_{cd}^*}\) at right far field(\({x=\infty}\)) and unique weak solution \({U}\) of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to \({{U_l}^{\pm}}\) and \({{U_r}^{\pm}}\) at \({x=-\infty}\) and \({x=\infty}\) respectively. For that purpose, we establish piecewise C1 estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of \({\partial\mathcal{N}_0}\) and \({\delta}\).

Uniqueness of integrable solutions to $${\nabla \zeta=G \zeta, \zeta|_\Gamma = 0}$$ for integrable tensor coefficients G and applications to elasticity

Abstract: Let \({\Omega \subset \mathbb{R}^{N}}\) be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary \({\partial\Omega}\). We show that the solution to the linear first-order system $$ abla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if \({G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}\) and \({\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}\). As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}( abla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for \({P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}\) with Curl \({P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}\), Curl \({P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}\) for some p, q > 1 with 1/p + 1/q = 1 as well as det \({P \geq c^+ > 0}\). We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let \({\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}\) satisfy sym \({( abla\Phi^\top abla\Psi) = 0}\) for some \({\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}\) with det \({ abla\Psi \geq c^+ > 0}\). Then, there exist a constant translation vector \({a \in \mathbb{R}^{3}}\) and a constant skew-symmetric matrix \({A \in \mathfrak{so}(3)}\), such that \({\Phi = A\Psi + a}\).

Non-uniform dependence on initial data for the μ − b equation

Abstract: This paper is concerned with the non-uniform dependence on initial data for the μ−b equation on the circle. Using the approximate solution method, we construct two solution sequences to show that the data-to-solution map of the Cauchy problem of the μ−b equation is not uniformly continuous in \({H^s(\mathbb{S})}\) .

On ground state solutions for superlinear Hamiltonian elliptic systems

Abstract: In this paper, we study the following Hamiltonian elliptic systems $$\left\{\begin{array}{ll}-\Delta u+V(x)u= g(x,v),\quad {\rm in }\, \mathbb{R}^N,\\-\Delta v+V(x)v= f(x,u),\quad {\rm in } \, \mathbb{R}^N.\end{array}\right.$$ where $${V(x)\in C(\mathbb R^N), f(x,t), g(x,t)\in C(\mathbb{R}^N\times \mathbb{R})}$$ are superlinear in t at infinity. Without Ambrosetti–Rabinowtitz condition, the existences of ground state solutions are obtained via the combination of generalized linking theorem and monotonicity method.

Some results for the primitive equations with physical boundary conditions

Abstract: In this paper, we consider the (simplified) 3-dimensional primitive equations with physical boundary conditions. We show that the equations with constant forcing have a bounded absorbing ball in the H1-norm and that a solution to the unforced equations has its H1-norm decay to 0. From this, we argue that there exists an invariant measure (on H1) for the equations under random kick-forcing.

Asymptotic behavior of second sound thermoelasticity with internal time-varying delay

Abstract: In this paper, we consider a thermoelastic system of second sound with internal time-varying delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Cattaneo’s law is still strong enough to uniformly stabilize the system even in the presence of time delay.

Closed-form solution for Eshelby’s elliptic inclusion in antiplane elasticity using complex variable

Abstract: This paper provides a closed-form solution for the Eshelby’s elliptic inclusion in antiplane elasticity. In the formulation, the prescribed eigenstarins are not only for the uniform distribution, but also for the linear form. After using the complex variable and the conformal mapping, the continuation condition for the traction and displacement along the interface in the physical plane can be reduced to a condition along the unit circle. The relevant complex potentials defined in the inclusion and the matrix can be separated from the continuation conditions of the traction and displacement along the interface. The expressions of the real strains and stresses in the inclusion from the assumed eigenstrains are presented. Results for the case of linear distribution of eigenstrain are first obtained in the paper.