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In present paper, we study the normalized solutions (λ c , u c) ∈ R × H 1 (R N) to the following Kirchhoff problem − (a + b ∫ R N | ∇ u | 2 d x) Δ u + λ u = g (u) in R N , 1 ≤ N ≤ 3 satisfying the normalization constraint ∫ R N u 2 = c , which appears in free vibrations of elastic strings. The parameters a , b > 0 are prescribed as is the mass c > 0. The nonlinearities g (s) considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given c > 0. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as c → 0 + as well as c → + ∞. [ABSTRACT FROM AUTHOR]

In the work, we consider the zero Mach number limit of compressible primitive equations in the domain R 2 × 2 T or T 2 × 2 T. We identify the limit equations to be the primitive equations with the incompressible condition. The convergence behaviors are studied in both R 2 × 2 T and T 2 × 2 T , respectively. This paper takes into account the high oscillating acoustic waves and is an extension of our previous work in [29]. [ABSTRACT FROM AUTHOR]

We deal with the following fractional Choquard equation (− Δ) s u + V (x) u = (I μ ⁎ | u | 2 μ , s ⁎ ) | u | 2 μ , s ⁎ − 2 u , x ∈ R N , where I μ (x) is the Riesz potential, s ∈ (0 , 1) , 2 s < N ≠ 4 s , 0 < μ < min ⁡ { N , 4 s } and 2 μ , s ⁎ = 2 N − μ N − 2 s is the fractional critical Hardy-Littlewood-Sobolev exponent. By combining variational methods and the Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this equation when V (x) is a positive potential bounded from below. The results obtained in this paper extend and improve some recent works in the case where the coefficient V (x) vanishes at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semi-hyperbolic patch characterized by 2D steady relativistic Euler equations. Employing the angle variables, the 2D steady relativistic Euler equations are transformed into a first-order hyperbolic equations. Given a smooth streamline and the boundary data, we find a C 1 , 1 6 -continuous sonic curve. Inside the semi-hyperbolic patch with the boundaries of the streamline associated with a characteristic curve, utilizing the partial hodograph method, a C 1 , 1 6 -continuous sonic-supersonic solution for 2D steady relativistic Euler equations is obtained. We will finally consider the corresponding regularity in the physical plane. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic solutions for some diffusive logistic equations. The main aim is to investigate the sharp different effects of local (Laplace) diffusion and nonlocal dispersal on the positive periodic solutions. Our result reveals that the spatial degeneracy plays different roles between reaction-diffusion equation and nonlocal dispersal equation. [ABSTRACT FROM AUTHOR]

We are concerned with the following constrained minimization problem: e (a 1 , a 2 , β) : = inf ⁡ { E a 1 , a 2 , β (u 1 , u 2) : ‖ u 1 ‖ L 2 (R 3) = ‖ u 2 ‖ L 2 (R 3) = 1 } , where E a 1 , a 2 , β is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters a 1 , a 2 , β and two trapping potentials V 1 (x) and V 2 (x). In this paper, we obtain the existence of minimizers of e (a 1 , a 2 , β) for possible a 1 , a 2 and β under suitable conditions on the potentials, which generalizes the results of the papers [17–19] in different senses. [ABSTRACT FROM AUTHOR]

In this paper, the averaging principle is studied for a class of multi-scale stochastic partial differential equations driven by α -stable process, where α ∈ (1 , 2). Using the technique of Poisson equation, we prove that the orders of strong and weak convergence are 1 − 1 / α and 1 − r for any r ∈ (0 , 1) respectively. The main contributions extend Wiener noise considered by Bréhier in [6] to α -stable process, and the finite dimensional case considered by Sun et al. in [36] to the infinite dimensional case. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to deal with the issue of well-posedness for a class of non-Newtonian fluid dynamics equations. The equations describing the motion of such fluids are characterized by a non-linear constitutive law relating the state of stress to the rate of deformation. We show the local-in-time existence and uniqueness of strong solutions to two important models: the Power Law model and the Bingham model. While our result for the first model holds over a periodic domain Ω = R 3 , the result obtained on the second model is limited to the one-dimensional case. This is because Bingham's constitutive law is discontinuous due to phase transition that may appear during the time when flows change nature, particularly from liquid motion to rigid motion and vice-versa. This property reduces the probability of showing smooth solutions to such a system in higher dimension space. [ABSTRACT FROM AUTHOR]

In the paper we established the L 2 estimates for commutators of the Dirichlet-to-Neumann map generated by a divergence form parabolic operator, defined on R + n + 2 , with real, symmetric and t , λ − independent coefficient matrix, or more generally, a small complex L ∞ perturbation of such. The major new challenge, compared to our previous work in elliptic setting, is to handle the first order derivatives with respect to the time variable t which not only fall on solutions to parabolic equations but also on auxiliary functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform L p − L q estimates with respect to a parameter β. From this one can cover the L p -boundedness of the Calderón-Zygmund operator with rough kernel by letting β → 0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space B p , q s and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following superlinear elliptic problem (P) { − Δ u = λ | u | p − 2 u + f (x , u) , in Ω , u = 0 , on ∂ Ω , where λ > 0 and 2 < p < 2 + δ for some δ > 0 small. The nonlinearity f satisfies the Ambrosetti-Rabinowitz condition and other appropriate hypotheses such that u = 0 is a local minimizer of the associated energy functional of equation (P). Our main novelties are threefold. Firstly, using the properties of Gromoll-Meyer pairs in Morse theory, we prove that equation (P) has at least one nontrivial solution close to 0. Moreover, four nontrivial solutions are obtained with assumptions on f at infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator −Δ. Therefore, our results differ significantly from those of the paper by Li and Li (2016) [16]. Secondly, under the assumptions of the paper above, we can obtain the existence of a fifth nontrivial solution of equation (P) for λ = 1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five nontrivial solutions of equation (P) based on the relationship between parameter λ and eigenvalues of operator −Δ. [ABSTRACT FROM AUTHOR]

In this paper, we study the prescribed Q -curvature flow equation on a arbitrary even dimensional closed Riemannian manifold (M , g) , which was introduced by S. Brendle in [3] , where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and ∫ M Q d μ < (n − 1) ! Vol (S n). In this paper we study the critical case that ∫ M Q d μ = (n − 1) ! Vol (S n) , we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence result in [21] in dimension 4 and extend the work of Li-Zhu [22] in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness problem for the generalized Benjamin-Bona-Mahony (gBBM) equation with quasi-periodic initial data on the real line. We obtain an existence and uniqueness result in the classical sense with arbitrary time horizon under the assumption of polynomially decaying initial Fourier data using the combinatorial analysis method developed in earlier papers by Christ [6] , Damanik-Goldstein [11] , and the present authors [12]. Our result is valid for exponentially decaying initial Fourier data and hence can be viewed as a Cauchy-Kovalevskaya theorem in the space variable for the gBBM equation with quasi-periodic initial data. [ABSTRACT FROM AUTHOR]

Consider the advective Fisher-KPP equation u t = u x x − β u x + f (u) on the half line [ 0 , ∞) with Dirichlet boundary condition at x = 0. In a recent paper [10] , the authors considered the problem without advection (i.e., β = 0) and constructed a new type of entire solution U (x , t) , which, under the additional assumption f ″ (u) ≤ 0 , is concave and U (∞ , t) = 1 for all t ∈ R. In this paper, we consider the equation with advection and without the additional assumption f ″ (u) ≤ 0. In case β = 0 , using a quite different approach from [10] we construct an entire solution U ˜ which is similar as U in the sense that U ˜ (∞ , t) ≡ 1 and U ˜ (⋅ , t) is asymptotically flat as t → − ∞ , but different from U in the sense that it does not have to be concave. Our result reveals that the asymptotically flat (as t → − ∞) property rather than the concavity is more essential for such entire solutions. In case β < 0 , we construct another new entire solution U ˆ which is completely different from the previous ones in the sense that U ˆ (∞ , t) increases from 0 to 1 as t increasing from −∞ to ∞. [ABSTRACT FROM AUTHOR]

We study the global existence of the parabolic-parabolic Keller–Segel system in R d , d ≥ 2. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter τ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler et al. (2015) [7] and Corrias et al. (2014) [12]. Our analysis improves earlier results and extends them to any dimension d ≥ 3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in τ , when τ ≫ 1 : we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller–Segel system. For these toy models, we establish in a companion paper [4] finite time blowup for a class of large solutions. [ABSTRACT FROM AUTHOR]

In this paper, we apply the quasi-reversibility method to solve an inverse source problem for a time-space fractional parabolic equation. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. The theoretical error estimates are confirmed with numerical tests for one and two dimensional equations. [ABSTRACT FROM AUTHOR]

This paper deals with a multi-objective control problem for the Kuramoto-Sivashinsky equation by following a Stackelberg-Nash strategy. We have a distributed control called Leader, and two boundary controls called Followers, each of them has to act over the equation to influence the behavior of the state in a particular way, by reaching or approaching to many targets at once. To be more precise, the Leader wants to drive the solution to a prescribed target at a final time, and the followers have to minimize some given cost functionals, adapting themselves to what the Leader wants. The main difficulty here is that, since the Followers are in the boundary, the problem turns to be equivalent to prove a partial null controllability result for a system of nonlinear fourth-order equations with boundary coupling terms. [ABSTRACT FROM AUTHOR]

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Gröbner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo–Mumford regularity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the incompressible limit of the compressible primitive equations with the effect of gravity. First we prove that the estimates of the local strong solutions to the compressible primitive equations with γ = 2 are uniform to the Mach number. Then we show that the local strong solutions of the system with well-prepared initial data, as well as their time derivatives, converge to those of the inhomogeneous incompressible primitive equations as the Mach number tends to zero. The convergence rates are shown to be identical to the Mach number. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the defocusing Schrödinger (NLS) equation with a nonzero background i q t + q x x − 2 (| q | 2 − 1) q = 0 , q (x , 0) = q 0 (x) , lim x → ± ∞ ⁡ q 0 (x) = ± 1. Recently, for the space-time region | x / (2 t) | < 1 which is a solitonic region without stationary phase points on the jump contour, Cuccagna and Jenkins presented the asymptotic stability of the N -soliton solutions for the NLS equation by using the ∂ ¯ generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form (0.1) q (x , t) = T (∞) − 2 q s o l , N (x , t) + O (t − 1) , whose leading term is N-soliton and the second term O (t − 1) is a residual error from a ∂ ‾ -equation. In this paper, we are interested in the large-time asymptotics in the space-time region | x / (2 t) | > 1 which is outside the soliton region, but there will be two stationary points appearing on the jump contour R. We found an asymptotic expansion that is different from (0.1) (0.2) q (x , t) = e − i α (∞) (1 + t − 1 / 2 h (x , t)) + O (t − 3 / 4) , whose leading term is a nonzero background, the second t − 1 / 2 order term is from the continuous spectrum and the third term O (t − 3 / 4) is a residual error from a ∂ ‾ -equation. The above two asymptotic results (0.1) and (0.2) imply that the region | x / (2 t) | < 1 considered by Cuccagna and Jenkins is a fast decaying soliton solution region, while the region | x / (2 t) | > 1 considered by us is a slow decaying nonzero background region. • We obtain large-time asymptotics in the solitonless region |x/(2t)| > 1 for defocusing NLS equation with nonzero background. • This result is a complete supplement to the result recently obtained by Cuccagna and Jenkins. • | x / (2 t) | < 1 considered by Cuccagna and Jenkins is a fast decaying soliton region, while ours | x / (2 t) | > 1 is a slow decaying nonzero background region. [ABSTRACT FROM AUTHOR]

This paper is concerned with a model of the two-dimensional zero thermal diffusivity magnetohydrodynamics-Boussinesq equations with the temperature-dependent viscosity and electrical conductivity. The main purpose of this paper is to establish the global regularity to this system with arbitrarily large initial data. [ABSTRACT FROM AUTHOR]

In this paper, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from u 0 is discontinuous at t = 0 in the metric of B 2 , ∞ s (R) , s > 3 2. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces B p , 1 1 + d p (R d) , 1 ≤ p < ∞ , which improves the local theory proved by [Zhou, Zhang & Mu, J. Differ. Equ. , 302(2021), pp.662-679]. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity ϵ Δ u as ϵ → 0 in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists (see Theorems 1.1 and 1.2). Furthermore, we prove the difference between two typical variational problems used to characterize the ground states (see Theorem 1.3). [ABSTRACT FROM AUTHOR]

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition G s for these ideals in terms of the heights of other ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the Rees rings of a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of the Rees ring in the generic case is an approximate resolution of the Rees ring in question. We end the paper by giving some explicit generation and concentration degree bounds. [ABSTRACT FROM AUTHOR]

The Sasa-Satsuma equation with 3 × 3 matrix spectral problem is one of the integrable extensions of the nonlinear Schrödinger equation. In this paper, we consider the Cauchy problem of the Sasa-Satsuma equation with generic decaying initial data. Based on the Rieamnn-Hilbert problem characterization for the Cauchy problem and the ∂ ‾ -nonlinear steepest descent method, we find qualitatively different long time asymptotic forms for the Sasa-Satsuma equation in three solitonic space-time regions: (1) For the region x < 0 , | x / t | = O (1) , the long time asymptotic is given by q (x , t) = u s o l (x , t | σ d (I)) + t − 1 / 2 h + O (t − 3 / 4) , in which the leading term is N (I) solitons, the second term the second t − 1 / 2 order term is soliton-radiation interactions and the third term is a residual error from a ∂ ‾ -equation. (2) For the region x > 0 , | x / t | = O (1) , the long time asymptotic is given by u (x , t) = u s o l (x , t | σ d (I)) + O (t − 1) , in which the leading term is N (I) solitons, the second term is a residual error from a ∂ ‾ -equation. (3) For the region | x / t 1 / 3 | = O (1) , the Painlevé asymptotic is found by u (x , t) = 1 t 1 / 3 u P (x t 1 / 3 ) + O (t 2 / (3 p) − 1 / 2) , 4 < p < ∞ , in which the leading term is a solution to a modified Painlevé II equation, the second term is a residual error from a ∂ ‾ -equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions to an initial boundary value problem of the Hunter-Saxton equation in the half space when the viscosity tends to zero. By means of the asymptotic analysis with multiple scales, we first formally derive the equations for boundary layer profiles. Next, we study the well-posedness of the equations for the boundary layer profiles by using the compactness argument. Moreover, we construct an accurate approximate solution and use the energy method to obtain the convergence results of the vanishing viscosity limit. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the large time behavior of the three-dimensional full compressible Navier-Stokes-Korteweg equations, which is used to model compressible viscous and heat-conductive fluids with internal capillarity, i.e., the liquid-vapor phase mixtures endowed with a variable internal capillarity. First, we construct the planar rarefaction wave to the three-dimensional full compressible Navier-Stokes-Korteweg equations, which can be derived by the fact that the rarefaction wave is nonlinearly stable to the one-dimensional full compressible Navier-Stokes-Korteweg equations. Then it is shown that the planar rarefaction wave is asymptotically stable provided that the initial data are a suitably small perturbation of the planar rarefaction wave and the strength of the rarefaction wave is small. The proof is based on the delicate energy method. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded domain (with smooth boundary) on the hyperbolic plane H n (1) , of center at origin and radius 1, in the (n + 1) -dimensional Lorentz-Minkowski space R 1 n + 1. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of k -convex solutions to a class of Hessian quotient equations defined over Ω ⊂ H n (1) and with the vanishing Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we develop new proof techniques and analytical methods to prove the existence of ground state solutions for the following planar Schrödinger-Poisson system with zero mass { − Δ u + ϕ u = f (u) , x ∈ R 2 , Δ ϕ = 2 π u 2 , x ∈ R 2 , where f ∈ C (R , R) has the critical exponential growth at infinity and there is no monotonicity restriction on f (u) / u 3. In particular, by using delicate estimates we obtain a desired upper bound for the Mountain Pass level just with the optimal asymptotic condition κ = lim inf | t | → ∞ t 2 F (t) e α 0 t 2 > 0 to restore the compactness in the presence of critical exponential growth, which significantly improves analogous assumptions on asymptotic behavior of t 2 F (t) e α 0 t 2 or t f (t) e α 0 t 2 at infinity in the previous works. Moreover, we use a different approach from the one of Du and Weth (2017) [21] dealing with the power nonlinearities to establish the Pohozaev type identity, which not only allows critical exponential growth nonlinearities, but also deals with the non-autonomous case containing a linear term V (x) u in the first equation, both of which are not covered in the existing literature. To our knowledge, there has not been any work in the literature on the subject, even for the simpler equation: − Δ u = f (u) in R 2. [ABSTRACT FROM AUTHOR]

In this paper, we consider the prescribed Q -curvature equation with a singularity Δ 2 u = | x | − α K (x) e 4 u in R 4 , Λ = ∫ R 4 | x | − α K (x) e 4 u d x < + ∞. First, we prove that all solutions are radially symmetric for 0 < α < 4 , K > 0 constant and Λ = 16 π 2 (1 − α 4). Next, some blow-up results have been studied by using a classification result for K > 0 constant and 0 < α < 4. Finally, we show that equation with K (x) = (1 − | x | p) and 0 < α < 4 has normal solutions (namely solutions which can be written in integral form) if and only if p ∈ (0 , 4 − α) and 8 π 2 (1 + p − α 4) ≤ Λ < 16 π 2 (1 − α 4). [ABSTRACT FROM AUTHOR]

For the famous Camassa-Holm equation, the well-posedness in B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞) and the ill-posedness in B p , r 1 + 1 p (R) with p ∈ [ 1 , ∞ ] , r ∈ (1 , ∞ ] had been studied in [13,14,16,23] , that is to say, it only left an open problem in the critical case B ∞ , 1 1 (R) proposed by Danchin in [13,14]. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in B ∞ , 1 1 (R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞ ] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u 0 ∈ B ∞ , 1 1 (R) but u 0 x 2 ∉ B ∞ , 1 0 (R) (an example implies B ∞ , 1 0 (R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u 0 x 2 ∈ B ∞ , 1 0 (R) holds, then the Camassa-Holm equation has a unique solution u (t , x) ∈ C T (B ∞ , 1 1 (R)) ∩ C T 1 (B ∞ , 1 0 (R)) and the norm inflation will not occur. [ABSTRACT FROM AUTHOR]

In this paper, we propose a unified framework for studying the Cahn–Hilliard equation with two distinct types of dynamic boundary conditions, namely, the Allen–Cahn and Cahn–Hilliard types. Using this unified framework, we develop a linear, second-order, and energy-stable scheme based on the multiple scalar auxiliary variables (MSAV) approach. We design efficient and decoupling algorithms for solving the corresponding linear system in which the unknown variables are intricately coupled both in the bulk and at the boundary. Several numerical experiments are shown to validate the proposed scheme, and to investigate the effect of different dynamical boundary conditions on the dynamics of phase evolution under different scenarios. [ABSTRACT FROM AUTHOR]

In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schrödinger operator, the proofs are based on the smooth perturbation theory by T. Kato. However, for the Klein-Gordon and Dirac equations, we also use a method of the microlocal analysis in order to prove the estimates for wider range of admissible pairs. As applications we prove the global existence of a solution to the higher order or fractional Hartree equation with potentials which describes the dynamics of infinitely many particles. We also give a local existence result for the semi-relativistic Hartree equation with electromagnetic potentials. As another application, the refined Strichartz estimates are proved for higher order and fractional Schrödinger, wave and Klein-Gordon equations. [ABSTRACT FROM AUTHOR]

This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order p 3 or the ten non-Abelian groups of order p 4 , p an odd prime, over a field of characteristic p. [ABSTRACT FROM AUTHOR]

The extended Fisher–Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biological systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence of the proposed scheme is proved rigorously. Based on the frame of the full-rank splitting scheme, we design a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve the energy dissipation. • The EFK equation is split into three subproblems, then a full-rank splitting scheme is established. The convergence of this scheme is analyzed. • A rank-adaptive low-rank approach is proposed for the EFK equation. To the best of our knowledge, this is new in the literature for the equation. • Numerical examples show that our methods are robust and accurate. They can also preserve energy dissipation. [ABSTRACT FROM AUTHOR]

Systems exhibiting Saddle-Node (SN) bifurcations are often characterized by drastic amplitude and phase jumps, representing a crucial state in engineering scenarios. The accurate and efficient prediction of SN points is fundamental for the comprehensive understanding and control of dynamical systems. This paper derives an equation for locating SN points based on the dimensionless governing equation for the in-plane primary resonance of a suspended cable. It reveals that the SN points for the cable are influenced by three key parameters: the cable's effective nonlinearity Γ e m , the excitation parameter F 2 , and damping ratio. Importantly, when the cable is subjected solely to horizontal end excitation, the product of Γ e m and F 2 emerges as a new parameter, Λ e m. The effects of parameter Λ e m (Γ e m), damping, axial, and vertical excitation amplitudes on SN points are investigated. Findings indicate that these key parameters more significantly affect the SN1 (the one near peck point) than SN2, and slight variations in Λ e m (Γ e m) or vertical excitation amplitude can lead to substantial alterations in SN1. The effect of Λ e m (Γ e m) on SN1 is asymmetric, with the values of σ and a being significantly higher when Λ e m (Γ e m) is positive than when negative. The computational results of the SN position equation closely align with experimental observations and the literature, demonstrating good computational efficiency. [ABSTRACT FROM AUTHOR]

Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes—seen as dynamical systems on a commutative ring—can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods. • We study the initialization of general lattice Boltzmann methods introducing an ad hoc modified equation analysis. • We find the constraints to obtain consistent initialization schemes, preserving second-order for the overall method. • We finely describe initial boundary layers due to dissipation mismatches between bulk and initialization schemes. • We introduce the observability of a lattice Boltzmann scheme, characterizing those with easily-mastered initializations. • We test the introduced analytical tools and their effectiveness through several—very conclusive—numerical experiments. [ABSTRACT FROM AUTHOR]

Efficient numerical schemes for solving the Poisson equation with jump conditions are of great interest for a variety of problems, including the modeling of electroporation phenomena and filamentary discharges. In this paper, we propose a modification to a finite volume scheme, namely the discrete dual finite volume method, in order to account for jump conditions with surface charges, i.e. with a source term. Our numerical tests demonstrate second-order convergence even with highly distorted meshes. We then apply the proposed method to model electroporation phenomena in biological cells by proposing a model that considers the thickness of the cell membrane as a separate domain, which differs from the literature. We show the advantages of the proposed method in this context through numerical experiments. • The discrete dual finite volume scheme is extended to solve the Poisson equation with jump conditions and surface charges. • The method is shown to exhibit a second-order convergence through canonical numerical tests. • The method is applied to the electroporation phenomena, where accurate modeling of the potential at the membrane is obtained. • Numerical experiments on the stationary and non-stationary case are provided. [ABSTRACT FROM AUTHOR]

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations. • Exponential Parareal notably reduces time-to-solution for non-diffusive equations. • Linear analysis accurately predicts Parareal performance on nonlinear problems. • Repartitioning is essential for stabilizing exponential integrators within Parareal. [ABSTRACT FROM AUTHOR]

The paper aims to study the geometry and physics of the Raychaudhuri equation (RE) in the background of a homogeneous and anisotropic space–time described by Kantowski–Sachs (KS) metric. Role of anisotropy/shear in the context of convergence and possible avoidance of singularity has been analyzed subject to a physically motivated constraint. Moreover, using a suitable transformation the first order RE has been converted to a second order differential equation analogous to a Harmonic Oscillator and criterion for convergence has been shown to be associated with the time varying frequency of the Oscillator. Finally, the paper points out a geometric and physical notion of anisotropy along with their corresponding behavior towards convergence. [ABSTRACT FROM AUTHOR]

The aim of this paper is to use the analytic theory of linear q -difference equations for the study of the functional-differential equation y ′ (x) = a y (q x) + b y (x) , where a and b are two non-zero real or complex numbers. When 0 < q < 1 and y (0) = 1 , the associated Cauchy problem admits a unique power series solution, ∑ n ≥ 0 (− a / b ; q) n n ! (b x) n , that converges in the whole complex x -plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis. [ABSTRACT FROM AUTHOR]

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L 2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori , each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field. • Semi-implicit FV/FE method for incompressible viscous and resistive MHD equations. • Well-balanced and exactly divergence-free on general unstructured mixed-element grids. • Constrained L2 projection for an exactly divergence-free reconstruction. • Thorough study of the lid-driven MHD cavity problem (reference solution is provided). • Stable long-time simulation of Grad-Shafranov equilibria in 3D tokamak geometries. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are (2 , 8) and (3 , 9). For triples, we prove the following theorem: If a ∉ { 2 , 8 } is a fixed integer larger than 1, then there are only finitely many triples (a , b , c) of pairwise distinct integers larger than 1 such that (a , b , c) , (a + 1 , b + 1 , c + 1) and (a + 2 , b + 2 , c + 2) are each multiplicatively dependent. Moreover, these triples can be determined effectively. [ABSTRACT FROM AUTHOR]

In this paper we obtain explicit estimates and existence results on the number of F q -rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number. [ABSTRACT FROM AUTHOR]

A Diophantine m -tuple with elements in the field K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K. Let X : (x 2 − 1) (y 2 − 1) (z 2 − 1) = k 2 , be an affine variety over K. Its K -rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x , y , z , k) ∈ X (K) is equal to k. We denote by X ‾ the projective closure of X and for a fixed k by X k a variety defined by the same equation as X. In this paper, we try to understand what can the geometry of varieties X k , X and X ‾ tell us about the arithmetic of Diophantine triples. First, we prove that the variety X ‾ is birational to P 3 which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a correspondence between a K3 surface X k for a given k ∈ F p × in the prime field F p of odd characteristic and an abelian surface which is a product of two elliptic curves E k × E k where E k : y 2 = x (k 2 (1 + k 2) 3 + 2 (1 + k 2) 2 x + x 2). We derive an explicit formula for N (p , k) , the number of Diophantine triples over F p with the product of elements equal to k. Moreover, we show that the variety X ‾ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X ‾ over an arbitrary finite field F q. Using it we reprove the formula for the number of Diophantine triples over F q from [DK21]. Curiously, from the interplay of the two (K3 and rational) fibrations of X ‾ , we derive the formula for the second moment of the elliptic surface E k (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S 4 (Γ 0 (8)). Finally, in the Appendix, Luka Lasić defines circular Diophantine m -tuples, and describes the parametrization of these sets. For m = 3 this method provides an elegant parametrization of Diophantine triples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness results of the three-dimensional incompressible Hall-magnetohydrodynamic equations with fractional dissipation. More precisely, we provide a direct proof of the local well-posedness of smooth solutions for the Hall-magnetohydrodynamic equations with the diffusive term for the magnetic field consisting of the fractional Laplacian with its power bigger than or equal to one half. Furthermore, the small data global well-posedness results are also derived. In addition, we obtain the optimal decay rate when the fractional powers are further restricted to a certain range. [ABSTRACT FROM AUTHOR]

We present the global existence and uniqueness of the classical solution to a system describing the interactions of short waves and long waves. This coupled system is formed by the compressible micropolar equations in Eulerian coordinates and the Schrödinger equation in Lagrangian coordinates. This paper establishes a sufficient frame condition to guarantee the well-posedness of the coupled system with large initial data in the two-dimensional torus. [ABSTRACT FROM AUTHOR]

We consider the Diophantine equation 7 x 2 + y 2 n = 4 z 3. We determine all solutions to this equation for n = 2 , 3 , 4 and 5. We formulate a Kraus type criterion for showing that the Diophantine equation 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper integer solutions for specific primes p > 7. We computationally verify the criterion for all primes 7 < p < 10 9 , p ≠ 13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x 2 + 7 y 2 n = 4 z 3 , determining all families of solutions for n = 2 and 3, as well as giving a (mostly) conjectural description of the solutions for n = 4 and primes n ≥ 5. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for Hartree type equation i u t + u x x = [ K ⁎ | u | 2 ] u , with Cauchy data in modulation spaces M p , q (R). We establish global well-posedness results in M p , p ′ (R) when K (x) = λ | x | γ , (λ ∈ R , 0 < γ < 1) with no smallness condition on initial data, where p ′ is the Hölder conjugate of p. Our proof uses a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Grünrock and Chaichenets et al. to the modulation space setting and exploits polynomial growth of the Schrödinger propagator on modulation spaces. [ABSTRACT FROM AUTHOR]

In this paper we study smoothness properties of solutions to a two-dimensional Kawahara equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ϕ possesses certain regularity and sufficient decay as x → ∞ , then the solution u (t) will be smoother than ϕ for 0 < t ≤ T where T is the existence time of the solution. [ABSTRACT FROM AUTHOR]