Related papers: Sharp local well-posedness and nonlinear smoothing…

We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser-Trudinger type inequalities and Strichartz estimates, we establish global well-posedness in the energy…

In this paper we prove optimal error estimates for {solutions with natural regularity} of the equations describing the unsteady motion of incompressible shear-thinning fluids. We consider a full space-time semi-implicit scheme for the…

We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids. We use the theory of operator semigroups in…

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma$ in $\mathbb R^d$, $d\ge 3$. Despite the simple geometric structure of such curves, the sharp…

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov regularizing effect, that means the smoothing…

We discuss local linear smooth backfitting for additive non-parametric models. This procedure is well known for achieving optimal convergence rates under appropriate smoothness conditions. In particular, it allows for the estimation of each…

The Cauchy problem for the derivative nonlinear Schr\"odinger equation with periodic boundary condition is considered. Local well-posedness for periodic initial data u_0 in the space ^H^s_r, defined by the norms ||u_0||_{^H^s_r}=||<xi>^s…

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for…

We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators $(-a^{ij}(x) \partial_{ij})^s$, for $0<s<1$, in bounded domains under minimal…

We analyze how the interaction between local and nonlocal dispersions, combined with different types of nonlinearities, influences the smoothing effects of solutions. To achieve this goal, we consider a model that generalizes the KdV and…

The nonlinear Schr\"odinger equation plays a fundamental role in mathematical physics, particularly in the study of quantum mechanics and Bose-Einstein condensation. This paper explores two distinct approaches to establishing the local…

We prove a sharp local existence result for the Schr\"odinger-Korteweg-de Vries system with initial data in $H^k(\mathbb{R})\times H^s(\mathbb{R})$. The proof is based on the concept of \textit{integrated-by-parts strong solution}, which…

We establish the local well-posedness of the Bartnik static metric extension problem for arbitrary Bartnik data that perturb that of any sphere in a Schwarzschild $\{t=0\}$ slice. Our result in particular includes spheres with arbitrary…

In this manuscript we establish local H\"older regularity estimates for bounded solutions of a certain class of doubly degenerate evolution PDEs. By making use of intrinsic scaling techniques and geometric tangential methods, we derive…

Given sufficiently regular data \textit{without} decay assumptions at infinity, we prove local well-posedness for non-linear dispersive equations of the form \[ \partial_t u + \mathsf A(\nabla) u + \mathcal Q(|u|^2) \cdot \nabla u= \mathcal…

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain up to some endpoints the full radial Strichartz estimates for the Schr\"odinger equation. The ideas of proof are…

We consider the Cauchy problem for the kinetic derivative nonlinear Schr\"odinger equation on the torus: \[ \partial_t u - i \partial_x^2 u = \alpha \partial_x \big( |u|^2 u \big) + \beta \partial_x \big[ H \big( |u|^2 \big) u \big] , \quad…

We establish sharp weighted smoothing estimates for limit solutions to the Cauchy-Dirichlet problem for the fast diffusion equation on smooth bounded domains. We demonstrate that the critical exponent governing these estimates coincides…

In the present paper, we consider the Cauchy problem of nonlinear Schr\"odinger equations with a derivative nonlinearity which depends only on $\bar{u}$. The well-posedness of the equation at the scaling subcritical regularity was proved by…

We study the derivative nonlinear wave equation \( - \partial_{tt} u + \Delta u = |\nabla u|^2 \) on \( \mathbb{R}^{1+3} \). The deterministic theory is determined by the Lorentz-critical regularity \( s_L = 2 \), and both local…