A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations

In dimension n?3, we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge-Ampère equation to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C^2,1 solutions, having n-1 nonvanishing principal curvatures, to certain subelliptic Monge-Ampère equations in dimension n?3. A corollary is that if k?0 vanishes only at nondegenerate critical points, then a C^2,1 convex solution u is smooth if and only if the symmetric function of degree n-1 of the principal curvatures of u is positive, and moreover, u fails to be when not smooth.     

Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities

In this paper, the author proves the global structure stability of the Lax's Riemann solution , containing only shocks and contact discontinuities, of general n×n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C¹ solution u=u(t,x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.     

Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes

In the first part of the paper, we prove the existence of a unique global solution to the axisymmetric Navier-Stokes system with initial data and external force with . This improves the result obtained by S. Leonardi, J. Málek, J. Necǎs and M. Pokorný [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649], where H²(R³) regularity was required. In the second part, we state global existence and uniqueness for the axisymmetric Navier-Stokes system with initial data in W2,p(R³) and external force in with 1<p<2. This also improves [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R³, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649] because less integrability is required on v₀ and on f.     

Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms

In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in R^N, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity.     

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.     

The Cauchy problem for the Schrödinger-KdV system

In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.     

Existence and regularity to an elliptic equation with logarithmic nonlinearity

We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in Ω⊂Rn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ^∗ such that uλ vanishes on a set of positive measure for 0<λ<λ^∗, and uλ>0 for λ>λ^∗. If f is concave, for λ>λ^∗ we characterize uλ by its stability.     

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies as t→∞, in , where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.     

Positivity properties in noncommutative convolution algebras with applications in pseudo-differential calculus

We study , of all such that for every ?∈C^∞₀, where denotes the twisted convolution. We prove that certain boundedness for are completely determined of the behaviour for a at origin, for example that , and that if a(0)<∞, then a∈L²∩L^∞. We use the results in order to determine wether positive pseudo-differential operators belong to certain Schatten-casses or not.     

Nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation

In was shown in Ruan et al. (2008) [3] that rarefaction waves for the generalized KdV-Burgers-Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small. The main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation. In our results, we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in , while for a general smooth nonlinear flux function, we need to ask for the L²-norm of the initial perturbation to be small but the L²-norm of the first derivative of the initial perturbation can be large and, consequently, the -norm of the initial perturbation can also be large.     

Existence of positive solutions for the Hénon equation involving critical Sobolev terms

Let N?3, ^2*=2N/(N−2) and Ω⊂RN be a bounded domain with a smooth boundary ∂Ω and 0∈Ω. Our purpose in this paper is to consider the existence of solutions of Hénon equation:

     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Existence and uniqueness of solutions for a non-uniformly parabolic equation

In this paper, we study a generalized Burgers equation u_t+(u²)_x=tu_xx, which is a non-uniformly parabolic equation for t>0. We show the existence and uniqueness of classical solutions to the initial-value problem of the generalized Burgers equation with rough initial data belonging to .     

Global regularity results for the 2D Boussinesq equations with vertical dissipation

This paper furthers the study of Adhikari et al. (2010) [2] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity v of any classical solution in the Lebesgue space Lq with 2?q<∞ is bounded by C₁q for C₁ independent of q. This bound significantly improves the previous exponential bound. In addition, we prove that, if v satisfies , then the associated solution of the 2D Boussinesq equations preserve its smoothness on [0,T]. In particular, implies global regularity.