Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameter α in the nonlinear term in the following parabolic equation■ It is shown that when 5/3 ≤ α < 7/3, the ■-dimensional parabolic Hausdorff measure of S is zero,which generalizes the recent corresponding work of Ozánski and Robinson in [SIAM J. Math. Anal.,51, 228–255(2019)] for α = 2 and f = 0. The same result is valid for a 3...     

Stability of Multidimensional Phase Transitions

In this paper, the author studies the multidimensional stability of subsonic phase transitions in a steady supersonic flow of van der Waals type. The viscosity capillarity criterion （in ＂Arch. Rat. Mech. Anal., 81（4）, 1983, 301-315＂） is used to seek physical admissible planar waves. By showing the Lopatinski determinant being non-zero, it is proved that subsonic phase transitions are uniformly stable in the sense of Majda （in ＂Mem. Amer. Math. Soc., 41（275）, 1983, 1-95＂） under both one dimensional and multidimensional perturbations.     

A revised pre-order principle and set-valued Ekeland variational principles with generalized distances

In my former paper "A pre-order principle and set-valued Ekeland variational principle"(see [J. Math. Anal. Appl., 419, 904–937(2014)]), we established a general pre-order principle.From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles(denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396(2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle,but also can be applied to obtain the above Khanh and Quy's EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.     

LOGARITHMICALLY IMPROVED REGULARITY CRITERION FOR THE 3D GENERALIZED MAGNETO-HYDRODYNAMIC EQUATIONS

This article proves the logarithmically improved Serrin's criterion for solutions of the 3D generalized magneto-hydrodynamic equations in terms of the gradient of the velocity field, which can be regarded as improvement of results in [10](Luo Y W. On the regularity of generalized MHD equations. J Math Anal Appl, 2010, 365: 806–808) and [18](Zhang Z J.Remarks on the regularity criteria for generalized MHD equations. J Math Anal Appl, 2011,375: 799–802).     

Generalized Gerstewitz's Functions and Vector Variational Principle for-Efficient Solutions in the Sense of Németh

In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle.J. Math. Anal. Appl., 419, 904–937(2014)]. By using the generalized Gerstewitz's functions and the partial order principle, we obtain a vector EVP for-efficient solutions in the sense of N′emeth, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.     

Global Classical Solutions to Partially Dissipative Quasilinear Hyperbolic Systems

The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems. Under the assumption that the system is weakly dissipative, Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in Arch. Rational Mech. Anal., Vol. 169, 2003, pp. 89–117). The aim of this paper is to give a completely different proof of this result with slightly different assumptions.     

Weak convergence theorem for Lipschizian pseudocontraction semigroups in Banach spaces

The purpose of this paper is to study the weak convergence problems of the irnplicity iteration process for Lipschitzian pseudocontraction semigroups in general Banach spaces. The results presented in this paper extend and improve the corresponding results of Zhou [Nonlinear Anal., 68, 2977-2983 （2008）], Chen, et ah [J. Math. Anal. Appl., 314, 701 709 （2006）], Xu and Ori [Numer. Funct. Anal. Optim, 22, 767-773 （2001）] and Osilike [J. Math. Anal. Appl., 294, 73-81 （2004）]. Keywords     

Energy-Preserving $H^1$--Galerkin Schemes for the Hunter-Saxton Equation

We consider the numerical integration of the Hunter–Saxton equation, which models the propagation of weakly nonlinear orientation waves. For the equation, we present two weak forms and their Galerkin discretizations. The Galerkin schemes preserve the Hamiltonian of the equation and can be implemented with cheap H~1 elements. Numerical experiments confirm the effectiveness of the schemes.     

Existence in the Large for Pressure-Gradient System

In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.     

New Kamenev-type oscillation criteria for half-linear partial differential equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping under quite general conditions. These results are extensions of the recent results developed by Sun [Y.G. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.     

Scattering Theory for the Defocusing Fourth Order NLS with Potentials

Based on the endpoint Strichartz estimates for the fourth order Schr?dinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schr?dinger operator. J. Funct. Anal., 274, 605–658(2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader,B.: The cubic fourth-order Schr?dinger equation. J. Funct. Anal., 256, 2473–2517(2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity 1 +8/n p 1 +8/(n-4) in dimensions n ≥ 7.     

CONVERGENCE OF AN IMMERSED INTERFACE UPWIND SCHEME FOR LINEAR ADVECTION EQUATIONS WITH PIECEWISE CONSTANT COEFFICIENTS Ⅱ: SOME RELATED BINOMIAL COEFFICIENT INEQUALITIES

In this paper we give proof of three binomial coefficient inequalities. These inequalities are key ingredients in [Wen and Jin, J. Comput. Math. 26, （2008）, 1-22] to establish the L^1-error estimates for the upwind difference scheme to the linear advection equations with a piecewise constant wave speed and a general interface condition, which were further used to establish the L^1-error estimates for a Hamiltonian-preserving scheme developed in [Jin and Wen, Commun. Math. Sci. 3, （2005）, 285-315] to the Liouville equation with piecewise constant potentials [Wen and Jin, SIAM J. Numer. Anal. 46, （2008）, 2688-2714].     

Stability of the Equilibrium to the Boltzmann Equation with Large Potential Force

The Boltzmann equation with external potential force exists a unique equilibrium—local Maxwellian. The author constructs the nonlinear stability of the equilibrium when the initial datum is a small perturbation of the local Maxwellian in the whole space R~3. Compared with the previous result [Ukai, S., Yang, T. and Zhao, H.-J.,Global solutions to the Boltzmann equation with external forces, Anal. Appl.(Singap.), 3,2005, 157–193], no smallness condition on the Sobolev norm H~1 of the potential is needed in our arguments. The proof is based on the entropy-energy inequality and the L~2-L~∞ estimates.     

SINGULAR AND RAREACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION

51. IntroductionIn this paper, we study the ealstence and regularity properties of weak solutions to thefonowillg nonlinear wave equationwhere c(.) is a given smooth, bounded, and positive function, "o(x) E Lip(B),ul(x) ELoo(R).Equation (1.1) is the Euler-Lagrange equation of the action principlefrom which comes the name variational wave equation. It is the simplest represelltative ofa large class of variational wave equatinns in classical field theories and general relativity(see [6]). We…     

SEGREGATED VECTOR SOLUTIONS FOR NONLINEAR SCHR?DINGER SYSTEMS IN R~2

We study the following nonlinear Schrodinger system where P(τ) and Q(τ) are positive radial functions,μ 0,v 0,and β∈ R is a coupling constant.This type of system arises,particularly,in models in Bose-Einstein condensates theory.Applying a finite reduction method,we construct an unbounded sequence of nonradial positive vector solutions of segregated type when β is in some suitable interval,which gives an answer to an interesting problem raised by Peng and Wang in Remark 4.1(Arch.Ration.Mech.Anal.,208(2013),305-339).     

On the Kernel of Restriction of Characters*

Let G be a finite group, H be a proper subgroup of G, and S be a unitary subring of C. The kernel of the restriction map S[Irr(G)] → S[Irr(H)] as a ring homomorphism is studied. As a corollary, the main result in [Isaacs, I. M. and Navarro, G., Injective restriction of characters, Arch. Math., 108, 2017, 437–439] is reproved.     

Mapping of Least ρ-Dirichlet Energy between Doubly Connected Riemann Surfaces

In this note, we consider the mappings h : X → Y between doubly connected Riemann surfaces having least ρ-Dirichlet energy. For a pair of doubly connected Riemann surfaces, in which X has finite conformal modulus, we establish the following principle: A mapping h in the class ■(X, Y)of strong limits of homeomorphisms in Sobolev space W1,2(X, Y) is ρ-energy-minimal if and only if its Hopf-differential is analytic in X and real along ?X. It improves and extends the result of Iwaniec et al.(see Theorem 1.4 in [Arch. Ration. Mech. Anal., 209, 401–453(2013)]). Furthermore, we give an application of the principle. Any ρ-energy minimal diffeomorphism is ρ-harmonic, however, we give a 1/|w|~2-harmonic diffemorphism which is not 1/|w|~2-energy minimal diffeomorphism. At last, we investigate the necessary and sufficient conditions for the existence of 1/|w|~2-harmonic mapping from doubly connected domain Ω to the circular annulus A(1, R).     

Existence results of solutions for anti-periodic fractional Langevin equation

Recently, Khalili and Yadollahzadeh [ Y. Khalili and M. Yadollahzadeh, Existence results for a new class of nonlinear Langevin equations of fractional orders, Iranian J. Sci. Tech., Trans. A: Sci., 2019, 43(5), 2335–2342] have investigated the uniqueness and existence of solution $u(t),~t\in[0,1]$ for a class of nonlocal boundary conditions to fractional Langevin equation. The authors used the boundary condition $u"(0)=0$ by incorrect method. In the current contribution, we show the correct method for using this condition and study the existence and uniqueness of solution for the same class of equation in slightly different form with anti-periodic and nonlocal integral boundary conditions as well as the boundary condition $u"(0)=0$. An exemplar is provided to illustrate our results.     

A Hardy–Trudinger–Moser Inequality Involving LpNorm in the Hyperbolic Space

Let B be the unit disc in R~2, H be the completion of C_0∞(B) under the norm■ .By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric ■, we prove that, for any fixed■ ,the supremum■ .This is an analog of early results of Lu–Yang(Discrete Contin. Dyn. Syst., 2009) and Yang(Trans.Amer. Math. Soc., 2007), and extends those of Wang–Ye(Adv. Math., 2012) and Yang–Zhu(Ann.Global Anal. Geom., 2016).