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It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of dilute electrons and plasma can be modeled by Vlasov–Poisson–Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is , by using a method developed for the Boltzmann equation without external force in [L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 (2005) 245–316]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro–micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components. [Copyright &y& Elsevier]

Abstract: Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine–Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47–84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257–289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101–132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large. [Copyright &y& Elsevier]

Abstract: This paper focuses on developing new Riccati type conditions for an abnormal time scale symplectic system (). These conditions provide characterizations of the nonnegativity (with and without a certain “image condition”) and positivity of the quadratic functionals associated with such a system. The novelty of these conditions rely on the natural conjoined basis of () in which is not necessarily invertible, and thus the system () could be abnormal. These results are new even in the special case of continuous time, as are some of them in the discrete time setting. [Copyright &y& Elsevier]

Abstract: Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree , with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly . The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l. [Copyright &y& Elsevier]

Abstract: The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with . The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n. We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions. [Copyright &y& Elsevier]

Abstract: In this paper, we study a free boundary problem for compressible spherically symmetric Navier–Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in -norm and weighted -norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of curved front solution of a partial differential equation coming from a mathematical model of stroke. The equation is of reaction–diffusion type in a cylinder of radius R and of diffusion and absorption type outside of the cylinder. We prove the nonexistence of a travelling front when R is small enough and the existence if R is large enough using a recent energy method. We construct the travelling front as the limit in time of a solution with a well-chosen initial condition, in a travelling referential. [Copyright &y& Elsevier]

Abstract: In this paper we study unique continuation properties of solutions to higher (fifth) order nonlinear dispersive models. The aim is to show that if the difference of two solutions of the equations, , decays sufficiently fast at infinity at two different times, then . [Copyright &y& Elsevier]

Abstract: Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface . If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones. [Copyright &y& Elsevier]

Abstract: In this paper, we study the biharmonic equation in the half-space , with . We prove in theory, with , existence and uniqueness results. We consider data and give solutions which live in weighted Sobolev spaces. [Copyright &y& Elsevier]

Abstract: We let Ω be a smooth bounded domain of and a sequence of functions such that in . We consider a sequence of functions such that in Ω for all . We address in this paper the question of the asymptotic behavior of the ''s when . The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the ''s are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. [Copyright &y& Elsevier]

Abstract: In this paper we consider nonlinear boundary value problems whose simplest model is the following: where is a summable function in Ω (bounded open set in , ), , and . [Copyright &y& Elsevier]

Abstract: The problem of a classification of robust homoclinic cycles in low-dimensional spaces has been frequently asked in recent years. In this paper, we resume the results in and and we solve the problem in in the case of orientation-preserving group actions. [Copyright &y& Elsevier]

This paper is concerned with a cross-diffusion system arising in a prey–predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions. [Copyright &y& Elsevier]

This paper investigates the effects of a degenerate diffusion term in reaction–diffusion models ut=[D(u)ux]x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c* and we show the appearance of a sharp-type profile when c=c*. These results solve recent conjectures formulated by Sa´nchez-Gardun˜o and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339). [Copyright &y& Elsevier]