Related papers: Well-posedness of a Debye type system endowed with…

We prove existence and uniqueness for the transport equation for currents (Geometric Transport Equation) when the driving vector field is time-dependent, Lipschitz in space and merely integrable in time. This extends previous work where…

We prove global well-posedness of the two-dimensional exterior Navier-Stokes equations for bounded initial data with a finite Dirichlet integral, subject to the non-slip boundary condition. As an application, we construct global solutions…

In this paper, we prove the local well-posedness of the Ericksen-Leslie system, and the global well-posednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the…

We first introduce a new model for a two-dimensional gauge-covariant wave equation with space-time white noise. In our main theorem, we obtain the probabilistic global well-posedness of this model in the Lorenz gauge. Furthermore, we prove…

In this paper, we study local well-posedness for the Navier-Stokes equations (NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces $\dot{H}^s_{L^{q, r}}(\mathbb{R}^d):= (-\Delta)^{-s/2}L^{q,r}$ for $d \geq 2, q > 1, s…

This paper presents a new approach to the local well-posedness of the $1d$ compressible Navier-Stokes systems with rough initial data. Our approach is based on establishing some smoothing and Lipschitz-type estimates for the $1d$ parabolic…

In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.

We show that first order semilinear PDEs by stochastic perturbation are well-posedness for globally Holder continuous and bounded vector field, with an integrability condition on the divergence. This result extends the liner case presented…

We consider the wave and Klein-Gordon equations on the real hyperbolic space $\mathbb{H}^{n}$ ($n \geq2$) in a framework based on weak-$L^{p}$ spaces. First, we establish dispersive estimates on Lorentz spaces in the context of…

We prove well-posedness for higher-order equations in the so-called dNLS hierarchy (also known as part of the Kaup-Newell hierarchy) in almost critical Fourier-Lebesgue and in modulation spaces. Leaning in on estimates proven by the author…

We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein--Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or…

In this paper, we consider the solvability of the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In [6], it was proved that the stationary Navier--Stokes equations on $\mathbb{R}^2$ is ill-posed for…

New local well-posedness results for dispersion generalized Benjamin-Ono equations on the torus are proved. The family of equations under consideration links the Benjamin-Ono and Korteweg-de Vries equation. For sufficiently high dispersion…

This paper is devoted to the well-posedness for dissipative KdV equations $u_t+u_{xxx}+|D_x|^{2\alpha}u+uu_x=0$, $0<\alpha\leq 1$. An optimal bilinear estimate is obtained in Bourgain's type spaces, which provides global well-posedness in…

We give a condition for the periodic, three dimensional, incompressible Navier-Stokes equations to be globally wellposed. This condition is not a smallness condition on the initial data, as the data is allowed to be arbitrarily large in the…

This paper is dedicated to the local existence theory of the Cauchy problem for a general class of symmetrizable hyperbolic partially diffusive systems (also called hyperbolic-parabolic systems) in the whole space $\mathbb{R}^d$ with $d\ge…

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.

We study forward and inverse problems for a semilinear radiative transport model where the absorption coefficient depends on the angular average of the transport solution. Our first result is the well-posedness theory for the transport…

In this paper, we prove the local well-posedness of 3-D axi-symmetric Navier-Stokes system with initial data in the critical Lebesgue spaces. We also obtain the global well-posedness result with small initial data. Furthermore, with the…

The well-posedness and regularity properties of diffusion-aggregation equations, emerging from interacting particle systems, are established on the whole space for bounded interaction force kernels by utilizing a compactness convergence…