Related papers: The classical Boussinesq system revisited

This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge2$) dimensional Boussinesq system is well-psoed for small initial data $(\vec{u}_0,\theta_0)$ ($\nabla\cdot\vec{u}_0=0$) either in…

In this article, we study the low-regularity Cauchy problem of a one dimensional quadratic Schrodinger system with coupled parameter $\alpha\in (0, 1)$. When $\frac{1}{2}<\alpha<1$,we prove the global well-posedness in $H^s(\mathbb{R})$…

This paper investigates solution stability properties of unregularized tracking-type optimal control problems constrained by the Boussinesq system. In our model, the controls may appear linearly and distributed in both of the equations that…

This paper provides a comprehensive Sobolev regularity theory for the Dirichlet problem of stochastic partial differential equations in $C^{1,\sigma}$ open sets. We consider substantially large classes of nonlocal operators and generalized…

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and…

We consider the compact case of one-dimensional quantum Zakharov system, as an initial-value problem with periodic boundary conditions. We apply the Bourgain norm method to show low regularity local well-posedness for a certain class of…

We address here the problem of regularity for weak solutions of the 3D Boussinesq equation. By introducing the new notion of partial suitable solutions, which imposes some conditions over the velocity field only, we show a local gain of…

The article studies the exact controllability and the stability of the sixth order Boussinesq equation \[ u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=f, \quad \beta=\pm1, \] on the interval $S:=[0,2\pi]$ with periodic boundary…

This paper aims to establish counterparts of fundamental regularity statements for solutions to elliptic equations in the setting of low-dimensional structures such as, for instance, glued manifolds or CW-complexes. The main result proves…

We address the persistence of regularity for the 2D $\alpha$-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for $(u_{0}, \rho_{0}) \in W^{s,q}(\mathbb R^2) \times…

We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schr\"odinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive…

We prove weighted Orlicz-Sobolev regularity for fully nonlinear elliptic equations with oblique boundary condition under asymptotic conditions of the following problem: $F(D^{2}u,Du,u,x)=f(x)$ in the bounded domain $\Omega\subset…

The paper deals with the regularity criterion for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution $(u,\theta )$ becomes regular provided that…

We study the Boltzmann equation near vacuum in anisotropic low-regularity Besov spaces. We establish the global existence and uniqueness of strong solutions with the critical regularity index $2/p$ for $p\in[1,\infty)$ in $\mathbb{R}^3$.…

In this paper, we study regularity of weak solutions to the incompressible Boussinesq equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of…

We study the three-dimensional Boussinesq system in bounded rough domains, including bounded Lipschitz and $\mathrm{C}^{1,\alpha}$ domains, within a critical functional framework. We establish existence and uniqueness results that are…

Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B'(u)$ is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation…

This paper explores the well-posedness of the Cauchy problem for the Fokker-Planck equation associated with the partial differential operator $L$ with low regularity condition. To address uniqueness, we apply a recently developed…

This work concerns stationary Stokes type systems governed by a general class of non-necessarily power-type nonlinearities. Fractional regularity properties of the symmetric gradient of local solutions are established, depending on a…

In this paper, we first address the space-time decay properties for higher order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a…