We show global well-posedness of the dynamic Φ4 model in the plane. The model is a non-linear stochastic PDE that can only be interpreted in a “renormalised” sense. Solutions take values in suitable weighted Besov spaces of negative regularity. MSC 2010: 81T27, 81T40, 60H15, 35K55.

We show that the Benjamin-Ono equation is globally well-posed in H s (R) for s ≥ 1. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in H s for any s [15]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.

Single view reconstruction approaches infer the structure of 3D objects or scenes from 2D images. This is an inherently ill-posed problem. An abundance of reconstruction approaches has been proposed in the literature, which can be characterized by the additional assumptions they impose to make the reconstruction feasible. These assumptions are either formulated by restrictions on the reconstruc...

The paper studies some inverse boundary value problem for simplest parabolic equations such that the homogenuous Cauchy condition is ill posed at initial time. Some regularity of the solution is established for a wide class of boundary value inputs.

We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space Ḣ1 0 . We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirich...

The sufficient conditions for unique local solvability, global solvability and of well-posedness of initial-boundary value problems for higher order nonlinear hyperbolic equations are studied.

We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equati...

We prove global well-posedness of the short-pulse equation with small initial data in Sobolev spaceHs for an integer s ≥ 2. Our analysis relies on local well-posedness results of Schäfer & Wayne [12], the correspondence of the short-pulse equation to the sine– Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also prove local and...

We prove that the Benjamin-Ono equation is globally well-posed in Hs(T) for s ≥ 0. Moreover we show that the associated flow-map is Lipschitz on every bounded set of Hs 0 (T), s ≥ 0, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on H∞ 0 (T)) cannot be of class C , α > 0, fr...

We present a local sensitivity analysis for the kinetic Cucker-Smale (C-S) equation with random inputs. This is a companion work to our previous local sensitivity analysis for the particle C-S model. Random imputs in the coefficients of the kinetic C-S equation can be caused by diverse sources such as the incomplete measurement and interactions with unknown environments, and will enter the prob...