One-dimensional reflected rough differential equations

We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. The whole difficulty of the problem (at least as far as uniqueness is concerned) lies in the non-continuity of the Skorohod map with respect to the topologies under consideration in the rough case. Our argument to overcome this obstacle is inspired by some ideas we introduced in a previous work dealing with rough kinetic PDEs arXiv:1604.00437.     

A type of time-symmetric forward–backward stochastic differential equations

In this Note, we study a type of time-symmetric forward–backward stochastic differential equations. Under some monotonicity assumptions, we establish the existence and uniqueness theorem by means of a method of continuation. We also give an application. To cite this article: S. Peng, Y. Shi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses

In this article, we investigate the orbital Hausdorff continuous dependence of the solutions to integer order and fractional nonlinear non-instantaneous differential equations. The concept of orbital Hausdorff continuous dependence is used to characterize the relations of solutions corresponding to the impulsive points and junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee this specific continuous dependence on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.     

Dynamic equations on time scales and generalized ordinary differential equations

The aim of this paper is to show that dynamic equations on time scales can be treated in the framework of generalized ordinary differential equations as introduced by J. Kurzweil. We also use some known results for generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales.     

Entropy solutions for stochastic porous media equations

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\mathrm{\Delta }(|u{|}^{m?1}u)$ for all $m\in (1,\infty )$, and Hölder continuous diffusion nonlinearity with exponent 1/2.     

Decay estimates for evolution equations with classical and fractional time-derivatives

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviours, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation.     

Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in $L_{loc}^1$ towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. As part of the proof, we also prove uniqueness of entropy solutions. We further provide a specific example of non-uniqueness of weak solutions and discuss the interplay of the entropy condition with the steady states. Finally, we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well-posed semigroup in the continuum limit, and showing the behaviour of solutions for large times.     

Nondegeneracy and stability of antiperiodic bound states for fractional nonlinear Schrödinger equations

We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schrödinger equations with fractional dispersion and power-law nonlinearity. As a key technical result, we demonstrate that the associated linearized operator is nondegenerate when restricted to antiperiodic perturbations, i.e. that its kernel is generated by the translational and gauge symmetries of the governing evolution equation. In the process, we provide a characterization of the antiperiodic ground state eigenfunctions for linear fractional Schrödinger operators on $\mathbb{R}$ with real-valued, periodic potentials as well as a Sturm–Liouville type oscillation theory for the higher antiperiodic eigenfunctions.     

Blowup with vorticity control for a 2D model of the Boussinesq equations

We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a vorticity stretching term and a non-local Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.     

Three sphere inequality for second order elliptic equations with coefficients with jump discontinuity

This is a short note to complete the paper appeared in Francini et al. (2016) [4], where a rough version of the classical well known Hadamard three-circle theorem for solution of an elliptic PDE in divergence form has been proved. Precisely, instead of circles, the authors obtain a similar inequality in a more complicated geometry. In this paper we clean the geometry and obtain a generalized version of the three-circle inequality for elliptic equation with coefficients with discontinuity of jump type.     

Density symmetries for a class of 2-D diffusions with applications to finance

We study densities of two-dimensional diffusion processes with one non-negative component. For such diffusions, the density may explode at the boundary, thus making a precise specification of the boundary condition in the corresponding forward Kolmogorov equation problematic. We overcome this by extending a classical symmetry result for densities of one-dimensional diffusions to our case, thereby reducing the study of forward equations with exploding boundary data to the study of a related backward equation with non-exploding boundary data. We also discuss applications of this symmetry for option pricing in stochastic volatility models and in stochastic short rate models.     

A minimax inequality and its applications to ordinary differential equations

The aim of this paper is to investigate the minimax inequality which plays a fundamental role in the critical points theorem of B. Ricceri below. Equivalent formulations are shown, and characterization is proved in particular for a special class of functionals. As an application, a multiplicity result for an ordinary Dirichlet problem is emphasized.     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

Singular solutions of elliptic equations on a perturbed cone

In this work we obtain positive singular solutions of

$\{\begin{array}{ccc}?\mathrm{\Delta }u(y)\hfill & \hfill =\hfill & u{(y)}^p\text{ in }y\in {\mathrm{\Omega }}_t,\hfill \\ u\hfill & \hfill =\hfill & 0\text{ on }y\in ?{\mathrm{\Omega }}_t,\hfill \end{array}$

where ${\mathrm{\Omega }}_t$ is a sufficiently small $C^{2,\alpha }$ perturbation of the cone $\mathrm{\Omega }:=\{x\in {\mathbb{R}}^N:x=r\theta ,r>0,\theta \in S\}$ where $S?S^{N?1}$ has a smooth nonempty boundary and where $p>1$ satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.     

Blow-up of solutions to critical semilinear wave equations with variable coefficients

We verify the critical case $p=p_0(n)$ of Strauss' conjecture [30] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in ${\mathbf{R}}^n$, where $n\ge 2$. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when $p=p_0(n)$. The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [43] and Zhou & Han [45]: exponential “eigenfunctions” of the Laplacian [37] are used to construct the test function ${?}_q$ for linear wave equation with variable coefficients and John's method of iterations [13] is augmented with the “slicing method” of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case.     

A generalized spline method for singular boundary value problems of ordinary differential equations

The boundary value problem for a linear first order system of ordinary differential equations with singularities at the endpoints of a finite interval is formulated. The convergence of a projection method involving generalized splines is investigated. The practical application of the method is discussed, and some numerical examples are presented.     

The optimal evolution of the free energy of interacting gases and its applications

We establish an inequality for the relative total – internal, potential and interactive – energy of two arbitrary probability densities, their Wasserstein distance, their barycenters and their generalized relative Fisher information. This inequality leads to many known and powerful geometric inequalities, as well as to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker–Planck type equations. It also yields the HWBI inequalities – which extend the HWI inequalities in Otto and Villani [J. Funct. Anal. 173 (2) (2000) 361–400], and in Carrillo et al. [Rev. Math. Iberoamericana (2003)], with the additional ‘B’ referring to the new barycentric term – from which most known Gaussian inequalities can be derived. To cite this article: M. Agueh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).