Related papers: A simple construction of the dynamical $\Phi^4_3$ …

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously…

We present a novel framework for the study of a large class of non-linear stochastic PDEs, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of…

The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and…

The primitive equations (PE) are a fundamental model in geophysical fluid dynamics. While the viscous PE are globally well-posed, their inviscid counterparts are known to be ill-posed. In this paper, we study the two-dimensional…

We present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each…

The motivation and the challenge in applying the renormalization group for systems with several scaling regimes is briefly outlined. The four dimensional $\phi^4$ model serves as an example where a nontrivial low energy scaling regime is…

We consider the defocusing fourth-order nonlinear Schr\"{o}dinger equation with potential \[ i\partial_t u + \Delta^2 u + Vu + \lambda |u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), \] in dimensions $n \ge 5$. In the…

We prove almost sure global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb{R}^3$ with random initial data in $ H^s(\mathbb{R}^3) \times H^{s-1}(\mathbb{R}^3)$ for $s > \frac 12$. The main new…

We analyze the stability properties of the so-called triple deck model, a classical refinement of the Prandtl equation to describe boundary layer separation. Combining the methodology introduced in [2], based on complex analysis tools, and…

We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the…

In this paper, we prove global well-posedness and scattering for the defocusing, cubic nonlinear Schr{\"o}dinger equation in three dimensions when $n = 3$ when $u_{0} \in H^{s}(\mathbf{R}^{3})$, $s > 3/4$. To this end, we utilize a…

We present a construction of the fractional $\Phi^4$ Euclidean quantum field theory on $\mathbb{R}^3$ in the full subcritical regime via parabolic stochastic quantisation. Our approach is based on the use of a truncated flow equation for…

We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain $\mathbb{T}^d$, $d\geq 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in…

In this paper, we study the well-posedness of the Forward-Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the…

We study the generalized derivative nonlinear Schr\"odinger equation $i\partial_t u+\Delta u = P(u,\overline{u},\partial_x u,\partial_x \overline{u})$, where $P$ is a polynomial, in Sobolev spaces. It turns out that when $\text{deg } P\geq…

This paper is devoted to the well-posedness of the inhomogeneous nonlinear wave equations. By combining Strichartz estimates with the contraction mapping principle, we establish local and global well-posedness in the function spaces…

It has recently been shown that the evolution of a linear Partial Differential Equation (PDE) can be more conveniently represented in terms of the evolution of a higher spatial derivative of the state. This higher spatial derivative (termed…

We prove that the $\Phi^4$ theory is trivial for any values of the bare coupling constant $\lambda$ thus extending previous results referring to very strong couplings to the full range of values for this parameter. The method is based on…

We prove the local well-posedness for the nonlinear fourth-order Schr\"odinger equation (NL4S) in Sobolev spaces. We also studied the regularity of solutions in the sub-critical case. A direct consequence of this regularity is the global…

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of…