On a multi-dimensional transport equation with nonlocal velocity

We study a multi-dimensional nonlocal active scalar equation of the form _ut+v⋅∇u=0$u_t+v\cdot \mathrm{\nabla }u=0$ in R⁺×R^d${\mathbb{R}}^{+}\times {\mathbb{R}}^d$, where v=Λ^−2+α∇u$v={\Lambda }^{-2+\alpha }\mathrm{\nabla }u$ with Λ=(−Δ)^1/2$\Lambda ={(-\mathrm{\Delta })}^{1/2}$. We show that when α∈(0,2]$\alpha \in (0,2]$ certain radial solutions develop gradient blowup in finite time. In the case when α=0$\alpha =0$, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.     

On the two-dimensional stationary Schrödinger equation with a singular potential

We study the equation

−△u(x,y)+ν(x,y)u(x,y)=0     

On the substantiation of a projection method for the stationary Fokker-Planck equation

A projection-type condition is discussed that is sufficient for the stationary Fokker-Planck equation Δu - div(u f) = 0 to be solvable within a class of probability density functions. Based on existence theorems and estimates of positive solutions obtained by the author, a fairly large class of vector fields f satisfying this condition is proposed.     

The inverse problem of determining the source in the stationary transport equation on a riemannian manifold

The transport equation with an unknown right-hand side is considered on a compact Riemannian manifold. The right-hand side of this equation is recovered from values of the outcoming flow. Assumptions under which the solution of the inverse problem is unique are formulated.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 236–242.     

On stationary distributions for the KPP equation with branching noise

This paper applies the method of Harris's convergence theorem for additive particle systems to a stochastic PDE that arises as the limit of long range contact processes. This is used to study the uniqueness of a translation invariant stationary distribution and its domain of attraction.

Résumé

On applique la méthode conduisant au théorème de convergence de Harris pour les systèmes de particules additifs à une EDP stochastique qui apparaît comme limite d'un processus de contact à longue portée. On étudie ainsi l'unicité de la mesure stationnaire invariante par translation et son domaine d'attraction.     

On the stability of the transport equation

We consider boundary value problems associated with the equation T=–A in a Hilbert Space, where T and A are bounded, self adjoint, injective, and A has a bounded inverse. We discuss the stability of the solution when A is perturbed by a self adjoint operator.     

Solution blow-up for a new stationary Sobolev-type equation

A new nonlinear stationary Sobolev-type equation with a parameter η ∈ ℝ¹ is derived. For η > 0, global solvability in the weak generalized sense is proved in the entire waveguide $ \mathbb{S} $ \mathbb{S} ⊗ ℝ₊¹. For η < 0, the strong generalized solution is shown to blow up in a certain waveguide cross section z = R ₀ > 0. An upper bound for R ₀ in terms of the original parameters of the problem is obtained.     

Solutions to a stationary nonlinear Black-Scholes type equation

We study by topological methods a nonlinear differential equation generalizing the Black-Scholes formula for an option pricing model with stochastic volatility. We prove the existence of at least a solution of the stationary Dirichlet problem applying an upper and lower solutions method. Moreover, we construct a solution by an iterative procedure.     

On the symmetry and exact solutions of a certain transport equation

Both the Lie andQ-conditional symmetry of a certain linear transport equation are studied and classes of its exact solutions are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 121–125, January, 1995.     

On bounded solutions of transport equation solved with a spectral method

In this article, a spectral method accompanied by finite difference method has been proposed for solving a boundary value problem that accompanies a stationary transport equation. We also prove that the solution is bounded by a value that depends of the source function. The accuracy and computational efficiency of the proposed method are verified with the help of a numerical example. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012