In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes γ-law, and the viscosity coefficient is a smooth function of density. The strength of shock wave could be arbitrarily large. The proof is given by introducing a new variable, which can formulate the original system into a new one, and the elementary energy method introduced in [21].

In this paper, we establish analyticity of solutions to the barotropic compressible Navier-Stokes equations describing the motion of the density ρ and the velocity field u in R3. We assume that ρ0 is a small perturbation of 1 and (1−1/ρ0,u0) are analytic in Besov spaces with analyticity radius ω>0. We show that the corresponding solutions are analytic globally in time when (1−1/ρ0,u0) are sufficiently small. To do this, we introduce the exponential operator e(ω−θ(t))D acting on (1−1/ρ,u), where D is the differential operator whose Fourier symbol is given by |ξ|1=|ξ1|+|ξ2|+|ξ3| and θ(t) is chosen to satisfy θ(t)<ω globally in time.

In this paper, we first introduce a new function space MHθ,p whose norm is given by the ℓp-sum of modulated Hθ-norms of a given function. In particular, when θ<−12, we show that the space MHθ,p agrees with the modulation space M2,p(R) on the real line and the Fourier-Lebesgue space FLp(T) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on R is globally well-posed in M2,p(R) for any p<∞, while the renormalized cubic NLS on T is globally well-posed in FLp(T) for any p<∞. In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.

In this paper we consider a spectral problem for ordinary differential equations of fourth order with the spectral parameter contained in three of the boundary conditions. We study the oscillatory properties of the eigenfunctions and, using these properties, we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space Lp(0,1),1

In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as η→∞.

In this paper, we obtain the boundary pointwise C1,α and C2,α regularity for viscosity solutions of fully nonlinear elliptic equations. That is, if ∂Ω is C1,α (or C2,α) at x0∈∂Ω, the solution is C1,α (or C2,α) at x0. Our results are new even for the Laplace equation. Moreover, our proofs are simple.

We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vector-valued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross-sectional area.

We show global existence and uniqueness of solutions for the 3D nonhomogeneous asymmetric fluids equations through a Lagrangian approach. In particular, uniqueness of the solution is proved under quite soft assumptions about its regularity, which brings the knowledge about nonhomogeneous asymmetric fluids to the same level as for the variable density Navier-Stokes equations.

We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity f(u) and coercive sign-changing potential V so that the Schrödinger operator −Δ+V is indefinite. Previous results in this framework required f to be strictly 3-superlinear, thus missing the paramount case of the Gross-Pitaevskii-Poisson system, where f(t)=|t|2t; in this paper we fill this gap, obtaining non-trivial solutions when f is not necessarily 3-superlinear.

Let La be a Schrödinger operator with inverse square potential a|x|−2 on Rn,n≥3. The main aim of this paper is to develop the theory of new Besov and Triebel–Lizorkin spaces associated to La based on the new space of distributions. As applications, we apply the theory to study some problems on the parabolic equation associated to La.

For Cr(r>1) random dynamical systems F on a compact smooth Riemannian manifold M, the fiber topological entropy is bounded above by an integral formula. Particularly, for the C∞ random dynamical systems, the integral formula coincides with the fiber topological entropy.

We construct a local Nekhoroshev-like result of stability with sharp constants for the planar three-body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted case.

In this paper, we go beyond what was proposed in theory by Melnikov ([15]) to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time-periodic equations. Not only we derive integral formula for splitting distances of all orders, but also we develop an analytic theory to evaluate the acquired multiple integrals. We reveal that the dominance of the exponentially small Poincaré/Melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order Melnikov integrals. This symmetry is beheld by many non-Hamiltonian equations.

We study the global dynamical behavior of Z2-equivariant cubic Hamiltonian vector fields with a linear type bi-center at (±1,0). By using a series of symbolic computation tools, we obtain all possible phase portraits of these Z2-equivariant Hamiltonian systems.

In this paper we consider a class of singularly perturbed domains, obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero. We use an Almgren-type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues, via Courant-Fischer Min-Max characterization and blow-up analysis for scaled eigenfunctions.

Let (Mn,gij) be a complete Riemannian manifold. We prove that for any p>n2, when k(p,1) is small enough, some parabolic type gradient bounds hold for the positive solutions of a nonlinear parabolic equationut=Δu+aulog⁡u, on geodesic balls B(O,r) in M with 0

In this paper we study the main properties of control sets with nonempty interior of linear systems on semisimple Lie groups. We show that, unlike the solvable case, linear systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.

We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navier slip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We prove the small-time global exact controllability of the position and velocity of the solid when the control takes the form of a distributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body. The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls for which the “Navier-Stokes+rigid body” system behaves like a perturbation of the “Euler+rigid body” system. By the means of a multi-scale asymptotic expansion we construct a controlled solution to the “Navier-Stokes+rigid body” system thanks to some controlled solutions to “Euler+rigid body”-type systems and by using that the influence of the boundary layer on the solid motion turns out to be sufficiently small.

We are concerned with the time-asymptotic behavior toward rarefaction waves for strong non-vacuum solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with degenerate density-dependent viscosity. The case when the pressure p(ρ)=ργ and the viscosity coefficient μ(ρ)=ρα for some parameters α,γ∈R is considered. For α≥0, γ≥max⁡{1,α}, if the initial data is assumed to be sufficiently regular, without vacuum and mass concentrations, we show that the Cauchy problem of the one-dimensional compressible Navier-Stokes equations admits a unique global strong non-vacuum solution, which tends to the rarefaction waves as time goes to infinity. Here both the initial perturbation and the strength of the rarefaction waves can be arbitrarily large. The proof is established via a delicate energy method and the key ingredient in our analysis is to derive the uniform-in-time positive lower and upper bounds on the specific volume.

In this paper, we study the hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations in a two or three dimensional bounded domain when the initial density is bounded away from zero. The proof relies on the relative entropy argument to obtain the strong convergence of macroscopic density of the particles nϵ in L∞(0,T;L1(Ω)), which extends the works of Goudon-Jabin-Vasseur [15] and Mellt-Vasseur [26] to inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations. Precisely, the relative entropy estimates in [15] and [26] give the strong convergence of uϵ and nϵ, ρϵ and nϵ, respectively. However, we only obtain the strong convergence of nϵ and uϵ from the relative entropy estimate, and we use another way to obtain the strong convergence of ρϵ via the convergence of uϵ. Furthermore, when the initial density may vanish, taking advantage of compactness result LM↪↪H−1 of Orlicz spaces in 2D, we obtain the convergence of nϵ in L∞(0,T;H−1(Ω)), which is used to obtain the relative entropy estimate, thus we also show the hydrodynamic limit for 2D inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations when there is initial vacuum.

Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation.

We study the optimal partitioning of a (possibly unbounded) interval of the real line into n subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as n tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function.

In this paper, we study two Liouville theorems for nonlinear boundary value problems on extended Grushin manifolds. These problems are closely related to nonlinear fractional Grushin equations. We will prove that these problems do not possess nontrivial positive solution under some assumptions on the nonlinear terms. We use the moving plane method in an integral form to prove our results.

In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.

We consider three types of semilinear equations (elliptic, parabolic and hyperbolic) posed in the N-dimensional exterior domain RN\D, where N≥2 and D is the closed unit ball in RN. A nontrivial Robin boundary condition is imposed on the boundary of D. Using a test function approach with judicious choices of the test functions, we show that the considered problems share a common critical behavior. We discuss separately the cases N=2 and N≥3. Moreover, in the case N≥3, the dependence of the critical exponent on initial data is discussed. To the best our knowledge, the study of the critical behavior in an exterior domain with a nontrivial Robin boundary condition has never been studied in the literature.

For the 3-dimensional steady irrotational isentropic supersonic flow against a general conic projectile, an improved explicit condition is obtained for the construction of approximate solution for the conical shock waves, and thereupon the existence of such conical shock solution is established under assumptions much weaker than previous ones.

The aim of the paper is to study a gradient constrained problem associated with a linear operator. Two types of problems are investigated. The first one is the equivalence between a non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. The equivalence result is obtained under a condition on the gradient constraint. The second problem is the existence of Lagrange multipliers. We prove that the non-constant gradient constrained problem admits a Lagrange multiplier, which is a Radon measure if the free term of the equation f∈Lp, p>1. If f is a positive constant, we regularize the result, namely we prove that the Lagrange multipliers belong to L2.

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Itô formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Itô formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random “noise”, under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process.

The purpose of this paper is to construct small-amplitude breather solutions for a nonlinear Klein-Gordon equation posed on a periodic metric graph via spatial dynamics and center manifold reduction. The major difficulty occurs from the irregularity of the solutions. The persistence of the approximately constructed pulse solutions under higher order perturbations is obtained by symmetry and reversibility arguments.

Let n≥2 and Ω be a bounded Lipschitz domain in Rn. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p∈(2,∞), two necessary and sufficient conditions for W1,p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W1,q estimates of solutions with q∈[2,p] and some Muckenhoupt weights, are obtained. As applications, for any given p∈(1,∞) and ω∈Ap(Rn) (the class of Muckenhoupt weights), the authors establish weighted Wω1,p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.

This paper focuses on investigating complete global topological classification of the three dimensional competitive Lotka-Volterra system with the identical intrinsic growth rate in the compactification of the positive octant of R3 including its infinity. The system is divided into 37 topological classes among 33 stable nullcline classes by the relation on the parameters of competitive system.

A well-posedness analysis of steady state of inviscid compressible subsonic impinging flows with non-zero vorticity is developed. A nonlinear nonhomogeneous second-order partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specifying the horizontal velocity. We first construct a subsonic solution to the impinging flow problem with special horizontal velocity and specific effluent fluxes in the outlets. Moreover, we showed that such a subsonic impinging flow with the same asymptotic behavior in the upstream and downstreams is unique. Furthermore, the existence, uniqueness and regularity of the interface separating the two fluids with different outlets are obtained. This result extends the recent result on the compressible subsonic irrotational impinging flows in [5]. Finally, as a byproduct, we obtain the existence of subsonic-sonic impinging flows.

We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small L2-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-Stokes equations.

In this paper, we study the following singularly perturbed Schrödinger-Poisson system{−ε2△u+V(x)u+ϕu=f(u)+u5,x∈R3,−ε2△ϕ=u2,x∈R3, where ε is a small positive parameter, V∈C(R3,R) and f∈C(R,R) satisfies neither the usual Ambrosetti-Rabinowitz type condition nor any monotonicity condition on f(u)/u3. By using some new techniques and subtle analysis, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0] the above system admits a semiclassical ground state solution vˆε with exponential decay at infinity. We also study the asymptotic behavior of {vˆε} as ε→0. In particular, our results can be applied to the nonlinearity f(u)∼|u|q−2u for q∈[3,4], and extend the previous work that only deals with the case in which f(u)∼|u|q−2u for q∈(4,6).

In this work, we study the stability of the expanding configurations of radiation gaseous stars. Such expanding configurations exist for a thermodynamic model, given as a class of self-similar solutions to the associated dynamic system with viscosity coefficients satisfying 2μ+3λ=0 for the monatomic gas; that is, the bulk viscosity is vanishing. With respect to small perturbations, this work shows that the linearly expanding homogeneous solutions are stable for a large expanding rate. This is an extensive study of the result in [14] by Hadžić and Jang.

We study the spectral theory for the first-order system Ju′+qu=wf of differential equations on the real interval (a,b) when J is a constant, invertible skew-Hermitian matrix and q and w are matrices whose entries are distributions of order zero with q Hermitian and w non-negative. Also, we do not pose the definiteness condition often required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval (a,b) are regular as well as the case of a 2×2 system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.

The aim of this paper is to show the maximal Lp-Lq regularity for a compressible fluid model of Korteweg type on general domains of the N-dimensional Euclidean space for N≥2 (e.g. bounded domains; exterior domains; half-spaces, layers, tubes, and their perturbed domains). Our approach is based on the theory of the R-boundedness for a generalized resolvent problem associated with the Korteweg-type model.

This paper is devoted to the study of the propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat. We first show that this equation admits a periodic forced wave with the speed at which the habitat is shifting by using the monotone iteration method combined with a pair of generalized super- and sub-solutions. Then we establish the nonexistence, uniqueness and global exponential stability of periodic forced waves by applying the sliding technique and the comparison argument. Finally, we obtain the spreading properties for a large class of solutions.

We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat n-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.

This paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time-scales, where the drift coefficients satisfy local Lipschitz conditions with respect to the slow and fast variables, and the coefficients in the slow equation depend on time t and ω. Making use of the techniques of time discretization and truncation, we prove that the slow component strongly converges to the solution of the corresponding averaged equation.

The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochastic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition.

This article is concerned with the isospectral problem−f″+14f=zωf+z2υf for the periodic conservative Camassa–Holm flow, where ω is a periodic real distribution in Hloc−1(R) and υ is a periodic non-negative Borel measure on R. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak⁎ topology.

We consider blow-up results for a system of inhomogeneous wave inequalities in exterior domains. We will handle three type boundary conditions: Dirichlet type, Neumann type and mixed boundary conditions. We use a unified approach to show the optimal criteria of Fujita type for each case. Our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation. We provide many new results and close some open questions.

In this paper we further explore the L-shadowing property defined in [20] for dynamical systems on compact spaces. We prove that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere satisfy this property. Homeomorphisms satisfying the L-shadowing property have a spectral decomposition where the basic sets are either expansive or contain arbitrarily small topological semi-horseshoes (periodic sets where the restriction is semiconjugate to a shift). To this end, we characterize the L-shadowing property using local stable and unstable sets and the classical shadowing property. We exhibit homeomorphisms with the L-shadowing property and arbitrarily small topological semi-horseshoes without periodic points. At the end, we show that positive finite-expansivity jointly with the shadowing property imply that the space is finite.

Inspired from the constructive method of Davilla et al. [10], with new ingredients, we extend their existence results to dimensions 7≤n≤9 concerning the following Hénon type problem{−Δu=K|u|p−1−εuinΩ,u=0on∂Ω, where Ω is a smooth bounded domain in Rn, ε is a positive real parameter, p+1=2n/(n−2) is the critical Sobolev exponent and the function K∈C2(Ω‾) is positive satisfying condition (1.1).

We consider the parabolic system ut−aΔu=f(v),vt−bΔv=g(u) in Ω×(0,T), where a,b>0, f,g:[0,∞)→[0,∞) are non-decreasing continuous functions and either Ω is a bounded domain with smooth boundary ∂Ω or the whole space RN. We characterize the functions f and g so that the system has a local solution for every initial data (u0,v0)∈Lr(Ω)×Ls(Ω), u0,v0≥0, r,s∈[1,∞).

We study a free boundary problem modeling solid tumor growth with ECM and MDE interactions. The production rate of MDE by tumor cells is a nonlinear function depending on nutrients and MDE. This nonlinear term is more complicated. For this model, we can not give the explicit expressions or integral expressions for MDE terms while we only get the existence and uniqueness. At first, we show the existence and the uniqueness of radially symmetric stationary solutions. Then the existence of symmetry-breaking solutions bifurcating from the radially symmetric stationary solutions is obtained.

We study the following system of equationsLi(ui)=Hi(u1,⋯,um)inRn, when m≥1, ui:Rn→R and H=(Hi)i=1m is a sequence of general nonlinearities. The nonlocal operator Li is given byLi(f(x)):=limϵ→0⁡∫Rn∖Bϵ(x)[f(x)−f(z)]Ji(z−x)dz, for a sequence of even, nonnegative and measurable jump kernels Ji. We prove a Poincaré inequality for stable solutions of the above system for a general jump kernel Ji. In particular, for the case of scalar equations, that is when m=1, it reads∬R2nAy(∇xu)[η2(x)+η2(x+y)]J(y)dxdy≤∬R2nBy(∇xu)[η(x)−η(x+y)]2J(y)dxdy, for any η∈Cc1(Rn) and for some nonnegative Ay(∇xu) and By(∇xu). This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in [46] for semilinear elliptic equations that is used extensively in the literature to establish De Giorgi type results, to study phase transitions and to prove regularity properties. We then apply this inequality to finite range jump processes and to jump processes with decays to prove De Giorgi type results in two dimensions. In addition, we show that whenever Hi(u)≥0 or ∑i=1muiHi(u)≤0 then Liouville theorems hold for each ui in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel Ji and a Liouville theorem for the quotient of partial derivatives of u.

In this article, we first prove, from the viewpoint of infinite dynamical system, sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. Then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a Liouville type equation. Moreover, we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the Liouville type equation. Finally, we study the equations of three-dimensional incompressible magneto-micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations.

We formulate a mathematical problem on hypersonic-limit of three-dimensional steady uniform non-isentropic compressible Euler flows of polytropic gases passing a straight cone with arbitrary cross-section and attacking angle, which is to study Radon measure solutions of a nonlinear hyperbolic system of conservation laws on the unit 2-sphere. The construction of a measure solution with density containing Dirac measures supported on the surface of the cone is reduced to find a regular periodic solution of highly nonlinear and singular ordinary differential equations (ODE). For a circular cone with zero attacking angle, we then proved the Newton's sine-squared law by obtaining such a measure solution. This provides a mathematical foundation for the Newton's theory of pressure distribution on three-dimensional bodies in hypersonic flows.

In this paper we give some Fujita type results for strongly p-coercive quasilinear parabolic differential inequalities with both a diffusion term and a dissipative term, whose prototype is given by ut−Δpu≥a(x)uq−b(x)um|∇u|s in RN×R+, u≥0, u(x,0)=u0(x)≥0 in RN, where p>1, q>0, 0≤m

In this paper we develop some results on Morse-Smale parabolic differential equation. We prove that the limiting problem of Morse-Smale parabolic equations is also Morse-Smale. The assumptions for obtaining this result are compact convergence of resolvent operators and the gap condition on eigenvalues. As application we exhibit a Morse-Smale Ordinary Differential Equation in RN, for each N≥2, and a scalar Morse-Smale semilinear equation with localized large diffusion.

In this paper, we show the existence of positive solutions for nonlinear Schrodinger equation with fractional Laplacian(−Δ)su+λu=|u|p−2uinΩ, where Ω⊂RN is an unbounded domain, ∂Ω≠∅ is bounded, λ∈R+, s∈(0,1), N>2s and 2

We consider a simplified compressible Navier-Stokes equations with cylindrical symmetry when viscosity coefficient λ and heat conductivity coefficient κ depend on temperature. We obtain global existence of strong solution and vanishing shear viscosity limit to the initial-boundary value problem in Eulerian coordinates. The analysis for the global existence is based on the assumption that μ=const.>0,1c˜θm≤λ(θ)≤c˜(1+θm),κ(θ)=θq, for m∈(0,1],q≥m. For the part of vanishing shear viscosity limit, we require in addition that 1c˜(1+θm)≤λ(θ)≤c˜(1+θm). In the paper, the acceleration effect in one direction is neglected, however, we do not need any smallness assumption for the initial data.

Based on a fixed point argument, we give a dynamical representation of the viscosity solution to Cauchy problem of certain weakly coupled systems of Hamilton-Jacobi equations with continuous initial datum. Using this formula, we obtain some regularity results related to the viscosity solution, including a partial extension of Lions' regularizing effect [17] to the case of weakly coupled systems.

We study the oscillation behavior of solutions to the heat equation on Rn and give some interesting examples. We compare the oscillation behavior of the initial data and the oscillation behavior of the solution as t→∞.

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of “very weak solutions” to the Cauchy problem, that was already successfully used in similar contexts in [12] and [20]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p-evolution equations for higher order operators on Rn or on groups, the results already being new in all these cases.

Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical approximations for the equation, gaining compactness properties. Together with the results in [30], we completely describe the blow-up phenomenon in case of uniformly bounded energy and zero weak limit in positive Yamabe class. In particular, for dimension greater or equal to five, Morse functions and with non-zero Laplacian at each critical point, we show that subsets of critical points with negative Laplacian are in one-to-one correspondence with such subcritical blowing-up solutions.

We study fully nonlinear elliptic equations with oblique boundary conditions. We obtain a global W2,p-estimate, n−τ0

We consider a system arising from a nonrelativistic Chern-Simon-Higgs model, in which a charged field is coupled with a gauge field. We prove an existence result for small coupling constants.