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Abstract: Abstract In this article, we consider the second-grade fluid equations in 2D exterior domain \(\Omega \) with homogeneous Dirichlet boundary conditions. For initial data \(\boldsymbol{u}_{0} \in \boldsymbol{H}^{3}(\Omega )\) , the second-grade fluid equations is shown to be globally well-posed. Furthermore, for arbitrary \(T > 0\) and \(s \geq 3\) , we prove that the solution belongs to \(L^{\infty}([0, T]; \boldsymbol{H}^{s}(\Omega ))\) provided that \(\boldsymbol{u}_{0}\) is in \(\boldsymbol{H}^{s}(\Omega )\) . PubDate: 2022-11-30

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Abstract: Abstract We consider the thermoelastic model following the type III theory for the Euler Bernoulli beam equation with tip. We prove that the corresponding semigroup is analytic. In particular, this implies: the smoothing effect over the initial data, the exponential stability of the semigroup and that the rate of decay of the semigroup is equal to the spectral bound of its generator (linear stability property). PubDate: 2022-11-29

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Abstract: Abstract In the \(p\) -wave spin-triplet pairing model of superfluid Helium-3, at each point \(x\) of the region \(\Omega \) occupied by the system, the order parameter field is described by a \(3\times 3\) complex matrix \(A\) encoding the orientation of the spin and orbital angular momentum of the Cooper pairs of Helium-3 atoms. The transition of liquid Helium-3 to a superfluid state is associated with a spontaneous breaking of the overall symmetry group \({\mathcal {G}}= SO(3) \times SO(3) \times U(1)\) of the system. In the Ginzburg–Landau regime (i.e., in regions near to the critical phase-transition temperature), the free-energy density \(f\) of superfluid Helium-3 is expanded into powers of the components \(A_{\mu j}\) of \(A\) and \(A_{\mu j, k}\) of its gradient \(\nabla A\) , and can be decomposed in the sum \(f(A, \nabla A) = f_{B}(A) + f_{\mathop{\mathrm{grad}}\nolimits }(A, \nabla A)\) of the bulk part, \(f_{B}\) , and the gradient part, \(f_{\mathop{\mathrm{grad}}\nolimits }\) . The free-energy density \(f\) must be invariant under the action of \({\mathcal {G}}\) defined by \(A \mapsto A' = \exp (i\phi ) R_{1}AR_{2}^{T}\) , where \(\phi \) is a phase, and \(R_{1}\) , \(R_{2}\) are elements of \(SO(3)\) , that is, it must be invariant against gauge transformations and against rotations in spin space and ordinary (orbital) space, separately. We address the question of \({\mathcal {G}}\) -invariance for a general free-energy density in the Ginzburg–Landau energy functional and determine all linearly independent quartic terms of the form \(AA^{*}\nabla A (\nabla A)^{*}\) in the expansion of the gradient free-energy density. It is known that the superfluid phases of Helium-3 near the critical temperature correspond to the minima of the bulk free energy and that the absolute minimum corresponds to a stable equilibrium phase. In zero magnetic field, there are two distinct superfluid phases, A and B, which exhibit an absolute minimum of the bulk free energy in different regions of the phase diagram. Explicit expressions for the generalized gradient energy densities are provided for both the A and B phases. Finally, a unified approach to A and B phases is proposed, which involves an auxiliary control parameter. In this framework, the extremal properties of A and B phases are recovered and a transition between the two phases is observed in dependence of pressure. PubDate: 2022-11-23

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Abstract: Abstract A perturbation theorem for regular sampling in the Paley-Wiener space, also known as the Kadec \(1/4\) -theorem, states that if \(\{x_{k}:k\in \mathbb{Z}\}\) is a sequence of real numbers for which \(L=\sup _{k\in \mathbb{Z}} x_{k}-k <1/4\) , then any entire function \(f\in L^{2}(\mathbb{R})\) of exponential type at most \(\pi \) can be recovered from its samples \(\{f(x_{k}):k\in \mathbb{Z}\}\) . Kadec-type theorems for irregular sampling in wavelet subspaces have been discussed in several papers. However, the optimal value of \(L\) is found only in the Franklin spline wavelet subspace. This paper aims to find a better bound for \(L\) in the Kadec-type theorem for wavelet subspaces along with sampling bounds. PubDate: 2022-11-23

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Abstract: Abstract This paper studies the existence and uniqueness of local weak solutions to the d-dimensional ( \(d\ge 2\) ) fractional micropolar Rayleigh-Bénard convection system without thermal diffusion. When the fractional dissipation index \(1\leq \alpha <1+\frac{d}{4}\) , any initial data \((u_{0},\omega _{0})\in B_{2,1}^{1+\frac{d}{2}-2\alpha}(\mathbb{R}^{d})\) and \(\theta _{0}\in B_{2,1}^{1+\frac{d}{2}-\alpha}(\mathbb{R}^{d})\) yield a local unique weak solution. PubDate: 2022-11-21

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Abstract: Abstract Existing literature shows that the strong ellipticity of an elasticity tensor can be equivalently transformed into the positive definiteness of even-order symmetric tensors constructed from the components of this elasticity tensor, and that an even-order symmetric tensor is positive definite if and only if all of its Z-eigenvalues are positive. In order to judge the strong ellipticity of an elasticity tensor, we first construct a Z-eigenvalue inclusion interval with parameters for an even-order symmetric tensor, and then by finding the optimal value of parameters we derive the optimal parameter interval, which is used to obtain a sufficient condition for the positive definiteness of the even-order symmetric tensor. Additionally, we prove that the new interval and sufficient condition are preferable than some existing ones. Finally, by using the positive definiteness of even-order symmetric tensors, we derived a necessary and sufficient condition and a checkable criterion for the strong ellipticity of an elasticity tensor. PubDate: 2022-11-16

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Abstract: Abstract Using variational tools combined with Nehari manifold approach, we discuss in this paper, the existence of solutions for a class of singular \(p(x)\) -biharmonic Laplacian problem with Navier boundary conditions. Precisely, we prove the existence of at least two nonnegative solutions for the given problem. PubDate: 2022-11-07

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Abstract: Abstract We present a complete algorithm for computing the affine equivalences between two implicit planar algebraic curves. We provide evidence of the efficiency of the algorithm, implemented in Maple, and compare its performance with existing algorithms. As a part of the process for developing the algorithm, we characterize planar algebraic curves, possibly singular, possibly reducible, invariant under infinitely many affine equivalences. PubDate: 2022-11-07

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Abstract: Abstract New \(Q\) -conditional (nonclassical) symmetries and exact solutions of the hunter-gatherer–farmer population model proposed by Aoki et al. (Theor. Popul. Biol. 50, 1–17 (1996)) are constructed. The main method used for the aforementioned purposes is an extension of the nonclassical method for system of partial differential equations. An analysis of properties of the exact solutions obtained and their biological interpretation are carried out. New results are compared with those derived in recent studies devoted to the same model. PubDate: 2022-11-07

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Abstract: Abstract This paper investigates the perturbed Hartree equation $$ i\dot{u}+\Delta u \pm u ^{2(p-1)}u\pm (I_{\alpha}* u ^{q}) u ^{q-2}u=0.$$ Indeed, one addresses the questions of global well-posedness and scattering versus finite time blow-up of energy solutions. One needs to deal with the absence of scaling invariance and try to understand the concurrency between a local and non-local source terms. The global existence and scattering of energy critical solutions for small data are proved regardless the sign of the source terms. In the case of two attractive non-linearity, the scattering of global solutions is proved in the inter-critical regime. Moreover, a decay result in available in the mass-sub-critical case. When there is an attractive part and a repulsive part in the source term, one gives a dichotomy of global existence versus finite time blow-up of solutions and the strong instability of standing waves, depending on a comparison between the exponents \(p\) and \(q\) . PubDate: 2022-10-19

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Abstract: Abstract As the first negative flow of the integrable generalization of the nonlinear Schrödinger equation, the Fokas-Lenells equation has attracted extensive attention in recent years. In this paper, we derive the general structure of the multi-component coupled Fokas-Lenells equations which have Lax representation in matrix form. Then we construct a basic theory of the general form of Lax pairs and Darboux transformations (classical and generalized) for the previously mentioned equation. As applications, we study two examples in detail, both of the four-component and the three-component coupled Fokas-Lenells equations can be reduced to the ubiquitous Fokas-Lenells equation. Furthermore, we apply the basic theory to obtain kinds of localized wave solutions, that is to say we use the classical Darboux transformation to obtain soliton solutions and use the generalized Darboux transformation to obtain soliton-positon solutions, rogue wave solutions and breather solutions. At last these localized wave solutions are illustrated by three-dimensional structure plots and two-dimensional density plots, as well as their dynamic properties are discussed. PubDate: 2022-10-14

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Abstract: Abstract The inverse problem of recovering a source term along with diffusion concentration for a generalized diffusion equation has been considered. The so-called 2nd level fractional derivative in time of order between 0 and 1 is used by fixing the parameters involved in 2nd level fractional derivative some well known fractional derivatives such as Riemann-Liouville, Caputo and Hilfer can be obtained. We investigated existence, uniqueness results and discussed some particular cases of the considered inverse problem. PubDate: 2022-10-11

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Abstract: Abstract Predators possess individual feeding behaviour in their territory. Mostly they attack prey according to their convenience and availability. Sometimes the predator predates upon more than one trophic level for survival. Also, the species go through self-interaction to acquire resources, habitats, food, mates, etc. In this study, we take a food web model consisting of basal prey, intermediate predator, and omnivorous as a top predator. We assume logistic growth for prey and intermediate predator. Here, the prey and the intermediate predator interaction is incorporated by the linear functional response, and other species interactions are followed by Holling Type II. We have studied the temporal as well as the spatio-temporal dynamics of this model. Turing instability conditions are also investigated. We generalize the effect of the diffusion coefficient on the spatial system. Higher-order stability analysis is explained with stability and instability criteria. Numerical verifications with the help of phase portraits, bifurcation, and Turing patterns are presented to illustrate the system’s dynamical behaviour and correlate with the applicability in the real world. We have implicated our model in the agricultural for omnivore carabid species to understand the consequence of their intraspecific interaction as well as trophic interactions. Also, we have validated this correlation with our model dynamics. PubDate: 2022-10-11

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Abstract: Abstract We consider the Cauchy problem for a system of fully nonlinear parabolic equations. In this paper, we shall show the existence of global-in-time solutions to the problem. Our condition to ensure the global existence is specific to the fully nonlinear parabolic system. PubDate: 2022-10-04

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Abstract: Abstract We study the asymptotic relations between certain singular and constrained control problems for one-dimensional diffusions with both discounted and ergodic objectives. In the constrained control problems the controlling is allowed only at independent Poisson arrival times. We show that when the underlying diffusion is recurrent, the solutions of the discounted problems converge in Abelian sense to those of their ergodic counterparts. Moreover, we show that the solutions of the constrained problems converge to those of their singular counterparts when the Poisson rate tends to infinity. We illustrate the results with drifted Brownian motion and Ornstein-Uhlenbeck process. PubDate: 2022-10-03

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Abstract: Abstract In this paper, we revisit a model for the contact line problem which has been proposed by Shikhmurzaev (Int. J. Multiph. Flow 19(4):589–610, 1993). In the first part, in addition to rederiving the model, we study in detail the assumptions required to obtain the isothermal limit of the model. We also derive in this paper several lubrication approximation models, based on Shikhmurzaev’s approach. The first two lubrication models describe thin film flow of incompressible fluids on solid substrates, based on different orders of magnitude of the slip length parameter. The third lubrication model describes a meniscus formation where a wedge-shaped solid immerses in a thin film of fluid. PubDate: 2022-09-28

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Abstract: Abstract A great deal of excellent work has been done for partial isometries. Thanks to early work of I. Erdélyi & P. R. Halmos, among others; they have played a fundamental role in structural study of Hilbert space operators, especially, in the theory of the polar decomposition of arbitrary operators and in the dimension theory of von Neumann algebras. They have also arisen in quantum physics (Bock et al. in Lett. Math. Phys. 112(2):1–11, 2022; Bracci and Picasso in Bull. Lond. Math. Soc. 39(5):792–802, 2007; Lai et al. in Quantum Inf. Process. 21(3):1–17, 2022). Based on the study of partial isometries (Erdélyi in J. Math. Anal. Appl. 22:546–551, 1968; Ezzahraoui et al. in Arch. Math. 110(3):251–259, 2018; Halmos and McLaughlin in Pac. J. Math. 13:585–596, 1963; Halmos and Wallen in J. Math. Mech. 19:657–663, 1970; Mostafa and Skhiri in Integral Equ. Oper. Theory 38:334–349, 2000; Wallen in Bull. Am. Math. Soc. 75:763–764, 1969) and semi-generalized partial isometries (Garbouj and Skhiri in Results Math. 75(1):15, 2020), for a given linear bounded non-zero operator \(A\) , we introduce a new class of operators called \(\mathcal{N}_{A}\) -isometries. We present its basic properties, and show a variety of results which improve and extend some works related to classical partial isometries. PubDate: 2022-09-27

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Abstract: Abstract We present a novel construction of recursion operators for integrable second-order multidimensional PDEs admitting isospectral scalar Lax pairs with Lax operators being first-order scalar differential operators linear in the spectral parameter. Our approach, illustrated by several examples and applicable to many other PDEs of the kind in question, employs an ansatz for the sought-for recursion operator of the equation under study based on the Lax pair for the latter. PubDate: 2022-09-20 DOI: 10.1007/s10440-022-00524-8

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Abstract: Abstract In this work, we investigate the existence and nonexistence of nonnegative solutions to a class of nonlocal elliptic systems set in a bounded open subset of , where the gradients of the unknowns act as source terms (see \((S)\) below). Our approach can be also used to treat other nonlinear systems with different structures. This work extends previous results obtained in the local case by the fourth author and his coworkers, and points to significant differences between the local and the nonlocal cases. PubDate: 2022-09-14 DOI: 10.1007/s10440-022-00528-4

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Abstract: Abstract In this paper, we establish the global existence of weak solutions for the incompressible Keller-Segel-Navier-Stokes equation with partial diffusion in . For the lack of diffusion \(\Delta \rho \) , we explore the structure of the equations and construct a priori estimates. With the help of uniform boundedness, we obtain the desired results. PubDate: 2022-09-12 DOI: 10.1007/s10440-022-00529-3