Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system

This paper is concerned with the existence, uniqueness, and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes–Korteweg system effected by the given mass source, the external force of general form, and the energy source in R³${\mathbb{R}}^3$. Based on the weighted L²$L^2$-method and some delicate L^∞$L^{\infty }$ estimates on solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier–Stokes–Korteweg system.     

KAM theory and the 3D Euler equation

We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold M   is not mixing in the C^k$C^k$ topology (k>4$k>4$ and non-integer) for any prescribed value of helicity and sufficiently large values of energy. This can be regarded as a 3D version of Nadirashvili's and Shnirelman's theorems showing the existence of wandering solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the mixing under the Euler flow of C^k$C^k$-neighborhoods of divergence-free vectorfields on M  . On the way we construct a family of functionals on the space of divergence-free C¹$C^1$ vectorfields on the manifold, which are integrals of motion of the 3D Euler equation. Given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types. We use the KAM theory to establish some continuity properties of these functionals in the C^k$C^k$-topology. This allows one to get a lower bound for the C^k$C^k$-distance between a divergence-free vectorfield (in particular, a steady solution) and a trajectory of the Euler flow.     

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.     

On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C²$C^2$ classical regularity, but also the existence of discontinuous entropy traveling wave solutions.     

Existence and optimal decay rates of the compressible non-isentropic Navier–Stokes–Poisson models with external forces

In this paper, we consider the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations with the potential external force. Under the smallness assumption of the external force in some Sobolev space, the existence of the stationary solution is established by solving a nonlinear elliptic system. Next, we show global well-posedness of the initial value problem for the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations, provided the prescribed initial data is close to the stationary solution. Finally, based on the elaborate energy estimates for the nonlinear system and L²$L^2$-decay estimates for the semigroup generated by the linearized equation, we give the optimal L²$L^2$-convergence rates of the solutions toward the stationary solution.     

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).     

On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N=2 or 3$N=2\text{or}3$. By exploiting a maximum principle, Nirenberg?s interpolation inequality and a smallness condition imposed on the N  -th component of initial direction field _d0${\mathbf{d}}_0$ to overcome the difficulties induced by the supercritical nonlinearity |∇d|²d${|\mathrm{\nabla }\mathbf{d}|}^2\mathbf{d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.     

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.     

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.     

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.     

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.     

Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. The existence of traveling wave solutions is obtained by the phase plane analysis, then the traveling wave solution is shown to be asymptotically stable by the elementary L²$L^2$-energy method.     

Physical measures and absolute continuity for one-dimensional center direction

For a class of partially hyperbolic C^k$C^k$, k>1$k>1$ diffeomorphisms with circle center leaves we prove the existence and finiteness of physical (or Sinai–Ruelle–Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions hold for an open and dense subset of all C^k$C^k$ partially hyperbolic skew-products on compact circle bundles.     

Existence of piecewise weak solutions of a discrete Cucker–Smale's flocking model with a singular communication weight

We prove existence of global C¹$C^1$ piecewise weak solutions for the discrete Cucker–Smale's flocking model with a non-Lipschitz communication weight ψ(s)=s^−α$\psi (s)=s^{-\alpha }$, 0<α<1$0<\alpha <1$. We also discuss the possibility of finite in time alignment of the velocities of the particles.     

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.     

Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions

This article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term. Both dissipation and the source are fully nonlinear   and the growth rate of the source map may include critical exponents (for Sobolev’s embedding H¹→L²$H^1\to L^2$). Besides continuity and monotonicity, no growth or regularity assumptions are imposed on the damping. We analyze the system in the presence of Neumann-type boundary conditions including the mixed cases: Dirichlet–Neumann–Robin.     

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$-x\cdot y$.     

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.