Electronic Research Archive
December 2021 , Volume 29 , Issue 6
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The semi-linear problem of a fractional diffusion equation with the Caputo-like counterpart of a hyper-Bessel differential is considered. The results on existence, uniqueness and regularity estimates (local well-posedness) of the solutions are established in the case of linear source and the source functions that satisfy the globally Lipschitz conditions. Moreover, we prove that the problem exists a unique positive solution. In addition, the unique continuation of solutions and a finite-time blow-up are proposed with the reaction terms are logarithmic functions.
A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the
In this paper, we introduce a simple local flux recovery for
In this paper, we show that for a nonsingular projective curve and a positive integer
In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of
This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.
In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted
The purpose of this paper is to study the constructions between Bihom-alternative superalgebras and Bihom-Malcev superalgebras and Bihom-Jordan superalgebras. First, we explain in detail that every regular Bihom-alternative superalgebra could be Bihom-Malcev-admissible superalgebra or Bihom-Jordan-admissible superalgebra. Next, the bimodules and
In this paper, we study the follwing important elliptic system which arises from the Lotka-Volterra ecological model in
Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball
We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.
This is a survey article for this special issue providing a review of the recent results in the mathematical analysis of active hydrodynamics. Both the incompressible and compressible models are discussed for the active liquid crystals in the Landau-de Gennes Q-tensor framework. The mathematical results on the weak solutions, regularity, and weak-strong uniqueness are presented for the incompressible flows. The global existence of weak solution to the compressible flows is recalled. Other related results on the inhomogeneous flows, incompressible limits, and stochastic analysis are also reviewed.
In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy
Using elementary methods, we count the quadratic residues of a prime number of the form
In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy.
We are concerned with the Cauchy problem of the 3D compressible Navier–Stokes–Poisson system. Compared to the previous related works, the main purpose of this paper is two–fold: First, we prove the optimal decay rates of the higher spatial derivatives of the solution. Second, we investigate the influences of the electric field on the qualitative behaviors of solution. More precisely, we show that the density and high frequency part of the momentum of the compressible Navier–Stokes–Poisson system have the same
This paper is devoted to the complete algebraic classification of complex
A classical stability criterion for Hill's equation is extended to more general families of periodic two-dimensional linear systems. The results are motivated by the study of mechanical vibrations with friction and periodic prey-predator systems.
In this paper, we consider the two-dimensional (2D) two-fluid boundary layer system, which is a hyperbolic-degenerate parabolic-elliptic coupling system derived from the compressible isentropic two-fluid flow equations with nonslip boundary condition for the velocity. The local existence and uniqueness is established in weighted Sobolev spaces under the monotonicity assumption on tangential velocity along normal direction based on a nonlinear energy method by employing a nonlinear cancelation technic introduced in [R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, J. Amer. Math. Soc., 28 (2015), 745-784; N. Masmoudi and T.K. Wong, Comm. Pure Appl. Math., 68(2015), 1683-1741] and developed in [C.-J. Liu, F. Xie and T. Yang, Comm. Pure Appl. Math., 72(2019), 63-121].
It is well-known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics with deformation. In particular, an elastic fluid strains when it is stretched and quickly returns to its original state once the stress is removed. In this review, we first introduce some mathematical results, which exhibit the stabilizing effect of elasticity on the motion of viscoelastic fluids. Then we further briefly introduce similar stabilizing effect in the elastic fluids.
In this paper, we consider the nonexistence problem for conformal Hessian quotient inequalities in
In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.
In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.
The objective of reinforcement learning (RL) is to find an optimal strategy for solving a dynamical control problem. Evolution strategy (ES) has been shown great promise in many challenging reinforcement learning (RL) tasks, where the underlying dynamical system is only accessible as a black box such that adjoint methods cannot be used. However, existing ES methods have two limitations that hinder its applicability in RL. First, most existing methods rely on Monte Carlo based gradient estimators to generate search directions. Due to low accuracy of Monte Carlo estimators, the RL training suffers from slow convergence and requires more iterations to reach the optimal solution. Second, the landscape of the reward function can be deceptive and may contain many local maxima, causing ES algorithms to prematurely converge and be unable to explore other parts of the parameter space with potentially greater rewards. In this work, we employ a Directional Gaussian Smoothing Evolutionary Strategy (DGS-ES) to accelerate RL training, which is well-suited to address these two challenges with its ability to (i) provide gradient estimates with high accuracy, and (ii) find nonlocal search direction which lays stress on large-scale variation of the reward function and disregards local fluctuation. Through several benchmark RL tasks demonstrated herein, we show that the DGS-ES method is highly scalable, possesses superior wall-clock time, and achieves competitive reward scores to other popular policy gradient and ES approaches.
This paper is concerned with the tempered pullback dynamics for a 3D modified Navier-Stokes equations with double time-delays, which includes delays on external force and convective terms respectively. Based on the property of monotone operator and some suitable hypotheses on the external forces, the existence and uniqueness of weak solutions can be shown in an appropriate functional Banach space. By using the energy equation technique and weak convergence method to achieve asymptotic compactness for the process, the existence of minimal family of pullback attractors has also been derived.
In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations
where the parameters
This article focuses on recent investigations on equilibria of the Frenkel-Kontorova models subjected to potentials generated by quasi-crystals.
We present a specific one-dimensional model with an explicit potential driven by the Fibonacci quasi-crystal. For a given positive number
A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.
In this survey we discuss old and new results on the number of critical points of solutions of the problem
In this paper, we consider the inviscid, incompressible planar flows in a bounded domain with a hole and construct stationary classical solutions with single vortex core, which is closed to the hole. This is carried out by constructing solutions to the following semilinear elliptic problem
We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to overcome a new difficulty arising from the treatment of boundary integrals due to the absence of the diffusion property for the velocity.
Ramanujan introduced sixth order mock theta functions
listed in the Lost Notebook. In this paper, we present some Ramanujan-like congruences and also find their infinite families modulo 12 for the coefficients of mock theta functions mentioned above.
This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.
This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop
under homogeneous Neumann boundary conditions in a bounded domain
The main theorem of this paper is that, for a general pair
We also easily show that, setting
is an embedding, then necessarily we have the inequality
Hence we formulate the following interesting conjecture, motivated by work of the second author: if
We prove the existence of regular optimal
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