Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C^1,1$C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C^1,1$C^{1,1}$ open set D$D$ and identify the Martin boundary of D$D$ with respect to the subordinate Brownian motion with the Euclidean boundary.     

On the two-phase membrane problem with coefficients below the Lipschitz threshold

We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C1,1$C^{1,1}$-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C¹$C^1$-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C1,1$C^{1,1}$-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points.     

Existence of global smooth solution to the initial boundary value problem for p-system with damping

This paper is concerned with the initial boundary value problem for the p$p$-system with damping. We prove the existence of the global smooth solution under the assumption that only the C⁰$C^0$-norm of the derivative of the initial data is sufficiently small, while the C⁰$C^0$-norm of the initial data is not necessarily small. The proof is based on several key a priori estimates, the maximum principle and the characteristic method.     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Energy decay for the linear Klein–Gordon equation and boundary control

In this work we study the asymptotic behavior of the solutions of the linear Klein–Gordon equation in R^N${\mathbb{R}}^N$, N?1$N?1$. We prove that local energy of solutions to the Cauchy problem decays polynomially. Afterwards, we use the local decay of energy to study exact boundary controllability for the linear Klein–Gordon equation in general bounded domains of R^N${\mathbb{R}}^N$, N?1$N?1$.     

Avoiding σ-porous sets in Hilbert spaces

We give a constructive proof that any σ  -porous subset of a Hilbert space has Lebesgue measure zero on typical C¹$C^1$ curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-p   porous sets may meet many C¹$C^1$ curves in positive measure.     

Approximate controllability for a system of Schrödinger equations modeling a single trapped ion

In this article, we analyze the approximate controllability properties for a system of Schrödinger equations modeling a single trapped ion. The control we use has a special form, which takes its origin from practical limitations. Our approach is based on the controllability of an approximate finite dimensional system for which one can design explicitly exact controls. We then justify the approximations which link up the complete and approximate systems. This yields approximate controllability results in the natural space (L²(R))²$(L^2{(\mathbb{R}))}^2$ and also in stronger spaces corresponding to the domains of powers of the harmonic operator.     

Boundary ε-regularity in optimal transportation

We develop an ε  -regularity theory at the boundary for a general class of Monge–Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on C²$C^2$ uniformly convex domains are C^1,α$C^{1,\alpha }$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost −x⋅y$-x\cdot y$.     

Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Rarefaction waves

In the present paper the author investigates the global structure stability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws under small BV perturbations of the initial data, where the Riemann solution contains rarefaction waves, while the perturbations are in BV but they are assumed to be C¹$C^1$-smooth, with bounded and possibly large C¹$C^1$-norms. Combining the techniques employed by Li–Kong with the modified Glimm’s functional, the author obtains a lower bound of the lifespan of the piecewise C¹$C^1$ solution to a class of generalized Riemann problems, which can be regarded as a small BV perturbation of the corresponding Riemann problem. This result is also applied to the system of traffic flow on a road network using the Aw–Rascle model.     

High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C¹$C^1$-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C¹$C^1$-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h⁴,k⁴)$O(h^4\text{,}{k}^4)$. Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C¹$C^1$-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.     

Insensitizing controls with one vanishing component for the Navier–Stokes system

In this paper we prove the existence of insensitizing controls, having one vanishing component, for the local L²$L^2$-norm of the solutions of the Navier–Stokes system. This problem can be recast as a null controllability problem for a nonlinear cascade system. We first prove a controllability result, with controls having one vanishing component, for a linear problem. Then, by means of an inverse mapping theorem, we deduce the controllability for the cascade system.     

On the inverse limit stability of endomorphisms

We present several results suggesting that the concept of C¹$C^1$-inverse (limit structural) stability is free of singularity theory. An example of a robustly transitive, C¹$C^1$-inverse stable endomorphism with a persistent critical set is given. We show that every C¹$C^1$-inverse stable, axiom A endomorphism satisfies a certain strong transversality condition (T). We prove that every attractor–repeller endomorphism satisfying axiom A and condition (T  ) is C¹$C^1$-inverse stable. The latter is applied to Hénon maps, rational functions and others. This leads us to conjecture that C¹$C^1$-inverse stable endomorphisms are exactly those which satisfy axiom A and condition (T).     

Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

We investigate global strong solution to a 3-dimensional parabolic–hyperbolic system arising from the Keller–Segel model. We establish the global well-posedness and asymptotic behavior in the energy functional setting. Precisely speaking, if the initial difference between cell density and its mean is small in L²$L^2$, and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H¹$H^1$, then they remain to be small in L²×H¹$L^2\times H^1$ for all time. Moreover, if the mean value of the initial cell density is smaller than some constant, then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity. The proof relies on an application of Fourier analysis to a linearized parabolic–hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration.     

Property of positive solutions and its applications for Sturm–Liouville singular boundary value problems

This paper is concerned with the property of the positive solutions for Sturm–Liouville singular boundary value problems with the linear conditions. We obtain a relation between the solutions and Green’s function. It implies a necessary condition for the C¹[0,1]$C^1[0,1]$ positive solutions. We apply the result to conclude that the given equation has no C¹[0,1]$C^1[0,1]$ positive solutions.     

Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications

We prove that the Schrödinger equation is approximately controllable in Sobolev spaces Hs$H^s$, s>0$s>0$, generically with respect to the potential. We give two applications of this result. First, in the case of one space dimension, combining our result with a local exact controllability property, we get the global exact controllability of the system in higher Sobolev spaces. Then we prove that the Schrödinger equation with a potential which has a random time-dependent amplitude admits at most one stationary measure on the unit sphere S   in L²$L^2$.     

To the theory of viscosity solutions for uniformly elliptic Isaacs equations

We show how a theorem about the solvability in C^1,1$C^{1,1}$ of special Isaacs equations can be used to obtain existence and uniqueness of viscosity solutions of general uniformly nondegenerate Isaacs equations. We apply it also to establish the C^1+χ$C^{1+\chi }$ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in C^γ$C^{\gamma }$ with γ slightly less than 1/2.     

Finite element boundary value integration of Wheeler–Feynman electrodynamics

The electromagnetic two-body problem is solved as a boundary value problem associated to an action functional. We show that the functional is Fréchet differentiable and that its conditions for criticality are the mixed-type neutral differential delay equations with state-dependent delay of Wheeler–Feynman electrodynamics. We construct a finite element method that finds C¹$C^1$-smooth solutions when suitable past and future positions of the particles are given as boundary data. The numerical trajectories satisfy a variational problem defined in a finite-dimensional Hermite functional space of C¹$C^1$ piecewise-polynomials. The numerical variational problem is solved using a combination of Newton’s method intercalated with boundary adjustments to ensure that the velocity of the solution is continuous with the boundary data. We recover the known circular orbits and compute several other novel trajectories of the Wheeler–Feynman electrodynamics. We also discuss the local convexity of the functional close to the new found trajectories and the possibility of solutions with less regularity.     

Exponential stability of nonlinear impulsive neutral integro-differential equations

In this paper, a nonlinear impulsive neutral integro-differential equation with time-varying delays is considered. By establishing a singular impulsive delay integro-differential inequality and transforming the n$n$-dimensional impulsive neutral integro-differential equation to a 2n$2n$-dimensional singular impulsive delay integro-differential equation, some sufficient conditions ensuring the global exponential stability in PC¹$P{C}^1$ of the zero solution of an impulsive neutral integro-differential equation are obtained. The results extend and improve the earlier publications. An example is also discussed to illustrate the efficiency of the obtained results.