Limits of nonlinear weak asymptotic methods

We propose a mathematical limit of $L^1$-stable weak asymptotic methods. A family of $L^1$-stable approximate solutions is transformed into a normal family of holomorphic functions defined in a complex domain having the real space on its boundary. This provides a holomorphic function which is the same mathematical object as the solutions from explicit calculations. The weak limit of the approximate solutions from weak asymptotic methods in the space of bounded Radon measures is recovered as a boundary value of this holomorphic function.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

On a free boundary problem for an American put option under the CEV process

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho =2r/{\sigma }^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.     

Blow-up solutions for a general class of the second-order differential equations on the half line

In this paper, we study the existence of positive blow-up solutions for a general class of the second-order differential equations and systems, which are positive radially symmetric solutions to many elliptic problems in ${\mathbb{R}}^N$. We explore fixed point arguments applied to suitable integral equations to get solutions.     

On the Boussinesq–Burgers equations driven by dynamic boundary conditions

We study the qualitative behavior of the Boussinesq–Burgers equations on a finite interval subject to the Dirichlet type dynamic boundary conditions. Assuming $H^1\times H^2$ initial data which are compatible with boundary conditions and utilizing energy methods, we show that under appropriate conditions on the dynamic boundary data, there exist unique global-in-time solutions to the initial-boundary value problem, and the solutions converge to the boundary data as time goes to infinity, regardless of the magnitude of the initial data.     

Strong solutions of the Boltzmann equation in one spatial dimension

For the Boltzmann equation, the setting of a narrow shock tube implies that solutions $f(x,v,t)$ depend upon $v\in {\mathbb{R}}^3$, however they have one-dimensional spatial dependence. This Note discusses the case in which solutions are periodic in x, with controlled total energy and entropy, and such that the macroscopic density determined by the initial data is bounded. Our principal result is that the macroscopic density then remains bounded at all subsequent times, that is, this data gives rise to strong solutions which exist globally in time. Through a weak/strong uniqueness principle, these solutions are unique among the class of dissipative solutions. Additionally, we show that the flow of the Boltzmann equation propagates the moments in $v\in {\mathbb{R}}^3$ and derivatives in both $x_1\in {\mathbb{R}}^1$ and $v\in {\mathbb{R}}^3$ of the solution $f(x,v,t)$. Our main theorems are valid for Boltzmann collision kernels which are bounded, and which have a relative velocity cutoff. The proofs depend upon a new averaging property of the collision operator and integral inequalities based in turn on entropy and on the Bony functional. To cite this article: A. Biryuk et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Random periodic solutions of SPDEs via integral equations and Wiener–Sobolev compact embedding

In this paper, we study the existence of random periodic solutions for semilinear SPDEs on a bounded domain with a smooth boundary. We identify them as the solutions of coupled forward–backward infinite horizon stochastic integral equations on $L^2(D)$ in general cases. For this we use Mercer?s Theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations. We then use the argument of the relative compactness of Wiener–Sobolev spaces in $C^0([0,T],L^2(\Omega \times D))$ and generalized Schauder?s fixed point theorem to prove the existence of a solution of the integral equations. This is the first paper in literature to study random periodic solutions of SPDEs. Our result is also new in finding semi-stable stationary solution for non-dissipative SPDEs, while in literature the classical method is to use the pull-back technique so researchers were only able to find stable stationary solutions for dissipative systems.     

Integrality and gauge dependence of Hennings TQFTs

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbb{k}$, starting from a finite Hopf algebra over $\mathbb{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds.We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotomic integers for surfaces with one boundary component.We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element.These results are combined and applied to show that the Hennings invariant associated to quantum-${\mathfrak{sl}}_2$ takes values in the cyclotomic integers. Using prior results of Chen et al. we infer integrality also of the Witten–Reshetikhin–Turaev $SO(3)$ invariant for rational homology spheres.As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras.     

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.