Weak Solutions for Stochastic FitzHugh–Nagumo Equations

This article studies the existence of weak solutions for a stochastic version of the FitzHugh–Nagumo equations. The random elements are introduced through initial values and forcing terms of associated Cauchy problem, which may be white noise in the time. Moreover there is a dependence of a stochastic parameter.     

Exact Solutions for Self-dual Yang–Mills Equations

The (constrained) canonical reduction of four dimensional self-dual Yang–Millstheory to 2, (2+1) dimensional sine-Gordon theory and 2 dimensional Liouvilles theory areconsidered. The Bäcklund transformations (BTs) areimplemented to obtain new classes of exact solutions for the reduced 2 dimensional sine-Gordonand Liouville models. Another transformation is developed and used to obtain exact solution forthe 2+1 and the original 3+1 sine-Gordon models.     

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.     

Optimal Three Cylinder Inequality at the Boundary for Solutions to Parabolic Equations and Unique Continuation Properties

Let Γ be a portion of a C ^1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D × (–T, T) and assume that u = 0 on Γ × (–T, T), 0 ∈ Γ. We prove that u satis.es a three cylinder inequality near Γ × (–T, T) . As a consequence of the previous result we prove that if u (x, t) = O (|x|^k) for every t ∈ (–T, T) and every k ∈ ℕ, then u is identically equal to zero. This work is partially supported by MURST, Grant No. MM01111258     

Global Solutions for the Two-Component Camassa–Holm System

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa–Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.     

Concentrations Problem for Solutions to Compressible Navier–Stokes Equations

Doklady Mathematics - A three-dimensional initial-boundary value problem for the isentropic equations of the dynamics of a viscous gas is considered. The concentration phenomenon is that, for...     

Mild Solutions for the Stochastic Generalized Burgers–Huxley Equation

Journal of Theoretical Probability - In this work, we consider the stochastic generalized Burgers–Huxley equation perturbed by space–time white noise and discuss the global solvability...     

Estimates for Solutions to Fokker–Planck–Kolmogorov Equations with Integrable Drifts

The result of this paper states that every probability measure satisfying the stationary Fokker–Planck–Kolmogorov equation obtained by a -integrable perturbation of the drift term–x of the Ornstein–Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure γ and \(f = \frac{{d\mu }}{{d\gamma }}\) for the density the integral of

$$f\left| {\log } \right|{\left( {f + 1} \right)^\alpha }$$

with respect to γ is estimated via \({\left\| v \right\|_{{L^1}\left( \mu \right)}}\) for all α < \(\frac{1}{4}\). This shows that stationary measures of infinite-dimensional diffusions whose drifts are integrable perturbations of–are absolutely continuous with respect to Gaussian measures. A generalization is obtained for equations on Riemannian manifolds.

     

Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces

Critical point theory is used to show the existence of weak solutions to a quasilinear elliptic differential equation under the functional framework of the Musielak–Sobolev spaces in a bounded smooth domain with Dirichlet boundary condition.     

Moment Propagation for Weak Solutions to the Vlasov–Poisson System

We consider weak solutions to the Cauchy problem for the three dimensional Vlasov–Poisson system of equations. We obtain a propagation result for any velocity moment of order > 2 as well as a uniqueness statement in ?³. In the periodic case, we show that velocity moments of order > 14/3 are propagated.     

Exact Solutions for Linearized Gravity in the Randall–Sundrum Model

We find exact solutions of the equations of motion for linearized gravity in the Randall–Sundrum model with matter on the branes and calculate the Newtonian limit of the model. The results established include the contributions of the radion and the massive gravitons, which essentially modify Newton's law at short distances. We consider the effects produced by shadow matter situated on the other brane and compare them with the effect of the usual matter for branes with positive and negative tension. We also calculate the deflection of light and Newton's law in the zero-mode approximation and explicitly isolate the contribution of the radion field.     

Multiple Solutions for Systems of Sturm–Liouville Boundary Value Problems

In this paper, the authors discuss the existence of multiple solutions to a class of second-order Sturm–Liouville boundary value systems. Their proofs are based on variational methods and critical point theory.     

Perturbation Method for Particle-like Solutions of the Einstein–Dirac Equations

The aim of this work is to prove by a perturbation method the existence of solutions of the coupled Einstein–Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. We relate the solutions of our equations to those of the nonlinear Choquard equation and we show that the nondegenerate solution of Choquard’s equation generates solutions of the Einstein–Dirac equations.     

Strong Unique Continuation Properties of Generalized Baouendi–Grushin Operators

This paper gives a quantitative control of the order of zero of a weak solution to perturbations of the Baouendi–Grushin operator, which generalizes the result due to Aronszaijn, Krzywicki, and Szarski valid for elliptic operators in divergence form with Lipschitz continuous coefficients.