Global existence and asymptotic convergence of weak solutions for the one-dimensional Navier-Stokes equations with capillarity and nonmonotonic pressure

We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v₀ is initially assumed to be in H¹ and the velocity variable u₀ to be in L² on a finite interval [0,1]. We show that both variables become smooth in positive time and that asymptotically in time u→0 strongly in L²([0,1]) and v approaches the set of stationary solutions in H¹([0,1]).     

Existence of three nontrivial solutions for an elliptic system

In this paper we consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones and computing the fixed point index in K₁, K₂ and K₁×K₂, we obtain that the elliptic system has three nontrivial solutions (u,0), (0,v) and (u^∗,v^∗). It is remarkable that the third nontrivial solution (u^∗,v^∗) is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better.     

HLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

We show the existence of dissipative H¨older continuous solutions of the Boussinesq equations. More precise, for any β∈(0,1/5), a time interval [0, T ] and any given smooth energy profile e : [0, T ] →(0, ∞), there exist a weak solution(v, θ) of the 3 d Boussinesq equations such that(v, θ) ∈ Cβ(T~3× [0, T ]) with e(t) =′his T~3|v(x, t)|~2 dx for all t ∈ [0, T ]. Textend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].     

Boundedness of global solutions for a porous medium system with moving localized sources

This paper deals with a class of porous medium systems with moving localized sources u_t=u^r₁(Δu+af(v(x₀(t),t))),v_t=v^r₂(Δv+bg(u(x₀(t),t))) with homogeneous Dirichlet boundary conditions. It is shown that under certain conditions, solutions of the above system blow up in finite time for large a and b or large initial data while there exist global positive solutions to the above system for small a and b or small initial data. Moreover, in the one dimensional space case, it is also shown that all global positive solutions of the above problem are uniformly bounded.     

L p -decay rates to nonlinear diffusion waves for p-system with nonlinear damping

In this paper, we study the L ^p (2 ? p ? +∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v(x,t), u(x,t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave (?(x,t), ū(x,t)) governed by the classical Darcys’s law provided that the corresponding prescribed initial error function (w ₀(x), z ₀(x)) lies in (H ³ × H ²) (?) and |v ₊ ? v _?| + ∥w ₀∥₃ + ∥z ₀∥₂ is sufficiently small. Furthermore, the L ^p (2 ? p ? +∞) convergence rates of the solutions are also obtained.     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

Stable viscosity matrices for systems of conservation laws

A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u₁ + f(u)_x = 0, u?R^m, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u_t + f(u)_x = v(Du_x)_x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u₁ + f′(u₀) u_x = vDu_xx should be well posed in L² uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

This paper deals with u_t = Δu + u^m(x, t)e^pv(0,t), v_t = Δv + u^q(0, t)e^nv(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u₀(v₀) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ? m ? 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions.     

Threshold and generic type I behaviors for a supercritical nonlinear heat equation

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|^p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume that N?3 and . Our first goal is to analyze a threshold behavior for solutions with initial data u₀=λv, where v∈C∩H¹ and v?0, v?0. It is known that there exists λ^?>0 such that the solution converges to 0 as t→∞ if 0<λ<λ^?, while it blows up in finite time if λ?λ^?. We show that there exist at most finitely many exceptional values λ₁=λ^?<λ₂λk such that, for all λ>λ^? with λ≠λj (j=1,2,…,k), the blow-up is complete and of type I with a flat local profile. Our method is based on a combination of the zero-number principle and energy estimates. In the second part of the paper, we employ the very same idea to show that the constant solution κ attains the smallest rescaled energy among all non-zero stationary solutions of the rescaled equation. Using this result, we derive a sharp criterion for no blow-up.     

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L^∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v₁=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.     

Quantitative unique continuation for the semilinear heat equation in a convex domain

In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation t_∂u−△u=g(u), with the homogeneous Dirichlet boundary condition, over Ω×(0,T_∗). Ω is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g:R→R satisfies certain conditions. We establish some observation estimates for (u−v), where u and v are two solutions to the above-mentioned equation. The observation is made over ω×{T}, where ω is any non-empty open subset of Ω, and T is a positive number such that both u and v exist on the interval [0,T]. At least two results can be derived from these estimates: (i) if ‖(u−v)(⋅,T)_‖L²(ω)=δ, then ‖(u−v)(⋅,T)_‖L²(Ω)?Cδα where constants C>0 and α∈(0,1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over ω×{T}, then they coincide over Ω×[0,Tm). Tm indicates the maximum number such that these two solutions exist on [0,Tm).     

On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations

Asymptotic properties of solutions of the nonlinear Klein-Gordon equation ?_t²u ? Δu + m²u + f(u) = 0 (NLKG) $\text{u¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{u¦}{}_0\text{= Ψ}$, are investigated, which are inherited from the corresponding solutions v of the (linear) Klein-Gordon equation ?_t²v ? Δv + m²v = 0$\text{v¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{v¦}{}_0\text{= Ψ}$, (KG) In particular, the finiteness of time-integrals in L_q over R₊ of certain Sobolevnorms in space of the solution is proved to be such a hereditary property. Together with a device by W. A. Strauss and a weak decay result for the (KG) due to R. S. Strichartz, this is used to prove that under suitable restrictions on the nonlinearity, the scattering operator for the (NLKG) is defined on all of L₂¹ × L₂ for n = 3.     

Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids

For any non-uniform lattice Γ in SL₂(?), we describe the limit distribution of orthogonal translates of a divergent geodesic in Γ\SL₂(?). As an application, for a quadratic form Q of signature (2, 1), a lattice Γ in its isometry group, and v ₀ ∈ ?³ with Q(v ₀) > 0, we compute the asymptotic (with a logarithmic error term) of the number of points in a discrete orbit v ₀Γ of norm at most T, when the stabilizer of v ₀ in Γ is finite. Our result in particular implies that for any non-zero integer d, the smoothed count for the number of integral binary quadratic forms with discriminant d ² and with coefficients bounded by T is asymptotic to c · T log T + O(T).     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations

In this paper, existence and attractiveness of solutions for quadratic Urysohn fractional integral equations on an unbounded interval are obtained by virtue of Tichonov fixed point theorem and suitable conjunction of the well known measure ω₀(X) and the spaces C(R⁺). Further, three certain solutions sets X_L,γ, X_1,α and X_{1,(1−(α+v))}, which tending to zero at an appropriate rate t^−ν (ν > 0), ν = γ (or α or 1 − (α + v)) as t → ∞, are introduced and stability of solutions for quadratic Urysohn fractional integral equations are obtained based on these solutions sets respectively by applying Schauder fixed point theorem via some easy checked conditions. An example is given to illustrate the results.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

Embedding long cycles in faulty k-ary 2-cubes

The class of k-ary n-cubes represents the most commonly used interconnection topology for distributed-memory parallel systems. Given an even k ? 4, let (V₁, V₂) be the bipartition of the k-ary 2-cube, f_v1, f_v2 be the numbers of faulty vertices in V₁ and V₂, respectively, and f_e be the number of faulty edges. In this paper, we prove that there exists a cycle of length k² − 2max{f_v1, f_v2} in the k-ary 2-cube with 0 ? f_v1 + f_v2 + f_e ? 2. This result is optimal with respect to the number of faults tolerated.     

On 2-designs

Denote by M_v the set of integers b for which there exists a 2-design (linear space) with v points and b lines. M_v is determined as accurately as possible. On one hand, it is shown for v > v₀ that M_v contains the interval [v + p + 1, v + p + q ? 1]. On the other hand for v of the form p² + p + 1 it is shown that the interval [v + 1, v + p ? 1] is disjoint from M_v; and if v > v₀ and p is of the form q² + q, then an additional interval [v + p + 1, v + p + q ? 1] is disjoint from M_v.