查看更多>>摘要：In this paper,we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation(-i▽-A(x))2u+λV(x)u=μ|u|p-2u+(∫RN|u(y)|2*α/|x-y|αdy)|u|2*α-2u in RN,(0.1)where N ≥ 4,2 ≤ p＜2*,2*α=2N-α/N-2 with 0＜α＜4,λ＞0,μ ∈ R,A(x)=(A1(x),A2(x),…,AN(x))is areal local Hölder continuous vector function,i is the imaginary unit,and V(x)is a real valued potential function on RN.Supposing that Ω=int V-1(0)⊂RN is bounded,we show that problem(0.1)possesses at least cat Ω(Ω)nontrivial solutions ifλ is large.
查看更多>>摘要：In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s2+|▽u|2)p-2/2▽u)=-div(|f|p-2f)+g in Q,u=h in ∂Ω,with the(sub-elliptic)degeneracy condition s ∈[0,1]and with mixed data f ∈ Lp(Ω;Rn),g ∈L/p-1(Q;Rn)for p ∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal dis-tribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of Mα and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of order α.Our approach therefore has its own interest.
查看更多>>摘要：The hydrodynamics of active liquid crystal models has attracted much attention in recent years due to many applications of these models.In this paper,we study the weak-strong uniqueness for the Leray-Hopf type weak solutions to the incompressible active liquid crystals in R3.Our results yield that if there exists a strong solution,then it is unique among the Leray-Hopf type weak solutions associated with the same initial data.
查看更多>>摘要：We study the incompressible limit of classical solutions to compressible ideal magneto-hydrodynamics in a domain with a flat boundary.The boundary condition is char-acteristic and the initial data is general.We first establish the uniform existence of classical solutions with respect to the Mach number.Then,we prove that the solutions converge to the solution of the incompressible MHD system.In particular,we obtain a stronger convergence result by using the dispersion of acoustic waves in the half space.
查看更多>>摘要：This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmo-spheric and oceanographic phenomena.The Boussinesq system focused on here is anisotropic,and involves only horizontal dissipation and thermal damping.In the 2D case R2,due to the lack of vertical dissipation,the stability and large-time behavior problems have remained open in a Sobolev setting.For the spatial domain T × R,this paper solves the stability problem and gives the precise large-time behavior of the perturbation.By decomposing the velocity u and temperature θ into the horizontal average(ū,(θ))and the corresponding oscilla-tion((u),(θ)),we can derive the global stability in H2 and the exponential decay of(ũ,(θ))to zero in H1.Moreover,we also obtain that(ū2,(θ))decays exponentially to zero in H1,and that ū1 decays exponentially to ū1(∞)in H1 as well;this reflects a strongly stratified phenomenon of buoyancy-driven fluids.In addition,we establish the global stability in H3 for the 3D case R3.
查看更多>>摘要：This paper is devoted to studying the stability of transonic shock solutions to the Euler-Poisson system in a one-dimensional nozzle of finite length.The background charge in the Poisson equation is a piecewise constant function.The structural stability of the steady transonic shock solution is obtained by the monotonicity argument.Furthermore,this transonic shock is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data.One of the crucial ingredients of the analysis is to establish the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions.
查看更多>>摘要：This paper is concerned with the global well-posedness of the solution to the com-pressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.
查看更多>>摘要：In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincaré map,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.
查看更多>>摘要：In this work,we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-Stewartson Ⅱ system(hyperbolic-elliptic case).Arbitrary order mass convergence could be achieved by the suitable addition of correction terms,while keeping the first order accuracy in Hγ × Hγ+1 for initial data in Hγ+1 × Hγ+1 with γ＞1.The main theorem is that,up to some fixed time T,there exist constants τo and C depending only on T and||u||L∞((o,T);Hγ+1)such that,for any 0＜τ ≤ To,we have that||u(tn,·)-un||Hγ ≤ Cτ,||v(tn,·)-vn||Hγ+1 ≤ Cτ,where un and vn denote the numerical solutions at tn=nτ.Moreover,the mass of the numerical solution M(un)satisfies that|M(un)-M(uo)|≤ Cτ5.
查看更多>>摘要：We study the existence and stability of monotone traveling wave solutions of Nicholson's blowflies equation with degenerate p-Laplacian diffusion.We prove the existence and nonexistence of non-decreasing smooth traveling wave solutions by phase plane analy-sis methods.Moreover,we show the existence and regularity of an original solution via a compactness analysis.Finally,we prove the stability and exponential convergence rate of traveling waves by an approximated weighted energy method.
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