Stability of plane Couette flows with respect to small periodic perturbations

We consider the plane Couette flow v₀=(x_n,0,…,0) in the infinite layer domain , where n≥2 is an integer. The exponential stability of v₀ in Lⁿ is shown under the condition that the initial perturbation is periodic in (x₁,…,x_n−1) and sufficiently small in the Lⁿ-norm.     

Energy-critical Hartree equation with harmonic potential for radial data

In this paper, we consider the defocusing, energy-critical Hartree equation with harmonic potential for the radial data in all dimensions (n≥5) and show the global well-posedness and scattering theory in the space Σ=H¹∩FH¹. We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well. Based on Bourgain and Tao’s approach, we use a localized Morawetz identity to show the global well-posedness. A key decay estimate comes from the linear part of the energy rather than the nonlinear part, which finally helps us to complete the scattering theory.     

Asymptotic dynamics of nonlinear Schrödinger equations with many bound states

We consider a nonlinear Schrödinger equation with a bounded localized potential in . The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order n in H¹, and that its ground state component is larger than n^3−ε with ε>0 small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity.     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

Local well-posedness for the homogeneous Euler equations

We introduce Triebel-Lizorkin-Lorentz function spaces, based on the Lorentz L^p,q-spaces instead of the standard L^p-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rⁿ,n≥2. As a corollary we obtain global existence of solutions to the 2D Euler equations in the Triebel-Lizorkin-Lorentz space. For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J.M. Bony.     

A removable isolated singularity theorem for the stationary Navier-Stokes equations

We show that an isolated singularity at the origin 0 of a smooth solution (u,p) of the stationary Navier-Stokes equations is removable if the velocity u satisfies u∈Lⁿ or |u(x)|=o(|x|^-1) as x→0. Here n?3 denotes the dimension. As a byproduct of the proof, we also obtain a new interior regularity theorem.     

Critical exponent for damped wave equations with nonlinear memory

We consider the Cauchy problem in Rⁿ,n≥1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t→∞ of small data solutions have been established in the case when 1≤n≤3. We also derive a blow-up result under some positive data in any dimensional space.     

On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω₀×(0,L)∈R³. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L_∞(0,L;L₂(Ω₀)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem.     

Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

Global and singular solutions to the generalized Proudman-Johnson equation

We show that there is a class of solutions to the generalized Proudman-Johnson equation which exist globally for all parameters a having the form for n∈N, thereby extending a result of Bressan and Constantin (2005) [2]. Furthermore, we present new proofs of existence of solutions developing spontaneous singularities and compute the corresponding blow-up rates.     

Optimal weighted estimates of the flows in exterior domains

In this paper, we prove some decay properties of global solutions for the Navier-Stokes equations in an exterior domain Ω⊂Rⁿ, n=2,3.When a domain has a boundary, the pressure term is troublesome since we do not have enough information on the pressure near the boundary. To overcome this difficulty, by multiplying a special form of test functions, we obtain an integral equation. He-Xin (2000) [12] first introduced this method and then Bae-Jin (2006, 2007) [1] and [13] modified their method to obtain better decay rates. Also, Bae-Roh (2009) [11] improved Bae-Jin’s results. Unfortunately, their results were not optimal, because there exists an unpleasant positive small δ in their rates.In this paper, we obtain the following optimal rate without δ,

     

The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains

The goal of this paper is to discuss the continuous dependence of solutions on functional parameters for the following semilinear elliptic partial differential equation: , for x∈Ω_r0?{x∈Rⁿ,n≥3,‖x‖>r₀} and v∈V, where V stands for some functional space. Our approach covers the case when f may change sign and admits general growth. As an additional result, the characterization of the radius r₀ for which our problem possesses at least one positive evanescent solution in the exterior domain Ω_r0 is described and numerically illustrated. Our approach relies on the subsolution and supersolution method and on a lemma due to Noussair and Swanson.     

New results of general n-dimensional incompressible Navier-Stokes equations

Let u=u(x,t,u₀) represent the global strong/weak solutions of the Cauchy problems for the general n-dimensional incompressible Navier-Stokes equations

     

Smoothing properties for a coupled system of nonlinear evolution dispersive equations

We study the smoothness properties of solutions to the coupled system of equations of Korteweg—de Vries type. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data (u₀, v₀ possesses certain regularity and sufficient decay as x → ∞, then the solution (u(t). v(t)) will be smoother than (u0, v0) for 0 < t ≤ T where T is the existence time of the solution.     

On global solution to the Klein-Gordon-Hartree equation below energy space

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R³. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S. Klainerman and D. Tataru, we establish the Hs (s<1) global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in Hs₀×Hs₀−1 where but not , for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation.     

Position dependent non-linear Schrödinger hierarchies: Involutivity, commutation relations, renormalisation and classical invariants

We consider a family of explicitly position dependent hierarchies , containing the NLS (non-linear Schrödinger) hierarchy. All are involutive and fulfill DIn=nIn−1, where D=D⁻¹V₀, V₀ being the Hamiltonian vector field afforded by the common ground state I₀=uv. The construction requires renormalisation of certain function parameters.It is shown that the ‘quantum space’ C[I₀,I₁,…] projects down to its classical counterpart C[p], with p=I₁/I₀, the momentum density. The quotient is the kernel of D. It is identified with classical semi-invariants for forms in two variables.     

Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices

A Hölder type inequality in Besov spaces is established and applied to show that every strong solution u(t,x) on (0,T) of the Navier-Stokes equations can be continued beyond t>T provided that the vorticity for 0<α<1.     

The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C¹ solutions with the bounded L¹∩L^∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C¹ travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R^1+(1+n).     

Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).