Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation

We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data \({(u_0,\theta_0)}\) are required to be only in the space \({X=\{f\in L^2(\mathbb{R}^2)\,|\,\partial_{x} f \in L^2(\mathbb{R}^2)\}}\), and thus our result generalizes that of Cao and Wu (Arch Rational Mech Anal, 208:985–1004, 2013), where the initial data are assumed to be in \({H^2(\mathbb{R}^2)}\). The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the \({L^\infty(\mathbb{R}^2)}\) norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.     

Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation

In this paper, we study the 2D Boussinesq equations with fractional Laplacian dissipation. In particular, we prove the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The main ingredient of the proof is the utilization of the Hölder estimates for advection fractional-diffusion equations as well as Littlewood–Paley technique.     

Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations

We study the initial boundary value problem of two-dimensional viscous Boussinesq equations over a bounded domain with smooth boundary. We show that the equations have a unique classical solution for H ³ initial data and the no-slip boundary condition. In addition, we show that the kinetic energy is uniformly bounded in time.     

A Derivation of the Volume-Averaged Boussinesq Equations for Flow in Porous Media with Viscous Dissipation

The volume-averaged equations are derived for convective flow in porous media. In the thermal energy equation viscous dissipation is taken into account, and a suitable form is obtained which is valid when Brinkman effects are significant.     

Note on Global Regularity for Two-Dimensional Oldroyd-B Fluids with Diffusive Stress

We prove global regularity of solutions of Oldroyd-B equations in two spatial dimensions with spatial diffusion of the polymeric stresses.     

Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRⁿ\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.     

Global Regularity for a Class of Generalized Magnetohydrodynamic Equations

It remains unknown whether or not smooth solutions of the 3D incompressible MHD equations can develop finite-time singularities. One major difficulty is due to the fact that the dissipation given by the Laplacian operator is insufficient to control the nonlinearity and for this reason the 3D MHD equations are sometimes regarded as “supercritical”. This paper presents a global regularity result for the generalized MHD equations with a class of hyperdissipation. This result is inspired by a recent work of Terence Tao on a generalized Navier–Stokes equations (T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equations, arXiv: 0906.3070v3 [math.AP] 20 June 2009), but the result for the MHD equations is not completely parallel to that for the Navier–Stokes equations. Besov space techniques are employed to establish the result for the MHD equations.     

Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations

This work is concerned with the partial regularity of the suitable weak solutions to the Boussinesq equations in \(\mathbb {R}^{n}\) where \(n=3,\,4\). By means of the De Giorgi iteration method developed in Vasseur (Nonlinear Differ Equ Appl 14(5–6):753–785, 2007), Wang, Wu (J Differ Equ 256(3):1224–1249, 2014), we obtain that \(n-2\) dimensional parabolic Hausdorff measure of the possible singular points set of the suitable weak solutions to this system is zero. Particularly, we obtain some interior regularity criteria only in terms of the scaled mixed norm of velocity for the suitable weak solutions to the Boussinesq equations, which implies that the potential singular points may only stem from the velocity field.     

Lipschitz Regularity for Elliptic Equations with Random Coefficients

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L ^∞-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (for example finite range of dependence). We also prove a quenched L ² estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.     

Solvability of the Boundary-Value Problems for the Boussinesq Equations with Inhomogeneous Boundary Conditions

The boundary-value problems for the stationary Boussinesq heat transfer equations with general non-standard boundary conditions for the velocity and mixed boundary conditions for the temperature are considered. The local and global existence theorems are proved. The precise a priori estimates for the solution are derived.     

An Accurate Finite Difference Scheme for Boussinesq Equations

In this paper, a stable and accurate finite difference scheme using a space-staggered grid is proposed for solving the extended Boussinesq-type equations as derived by Nwogu [Journal of Waterway, Port Coastal and Ocean Engineering, ASCE, 119, (1993) 618–638]. The alternate direction iterative method combined with an efficient predictor–corrector scheme is adopted for the numerical solution of the governing differential equations. The proposed method is verified by two test cases where experimental data are available for comparison. The first case is wave focusing by bottom topography as studied by Whalin [The limit of applicability of linear wave refraction theory in a convergence zone. Res. Rep. H-71-3, U.S.Army Corps of Engrs. Waterways Expt. Station, Vicksburg (1971)]. The second case is wave runup around a circular cylinder as investigated experimentally by Isaacson (Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 104, (1978), 69–79). Numerical results agree very well with the corresponding experimental data in both cases.     

Convergence of Global and Bounded Solutions of Some Nonautonomous Second Order Evolution Equations with Nonlinear Dissipation

In this paper we study the asymptotic behavior of solutions of the following nonautonomous wave equation with nonlinear dissipation.

$\left\{\begin{array}{ll} u_{tt}+\vert u_{t}\vert^{\alpha}u_{t}-\Delta u +f(u)=g(t,x),\quad{\rm in}\,\mathbb{R}_{+}\times\Omega,\\ \qquad\qquad u(t,x)=0,\quad\, {\rm on}\,\mathbb{R}_{+}\times\partial\Omega,\end{array}\right.$

where f is an analytic function, α is a small positive real and g(t, ·) tends to 0 sufficiently fast in L ²(Ω) as t tends to ∞.

We also obtain a general convergence result and the rate of decay of solutions for a class of second order ODE containing as a special case

$\left\{\begin{array}{ll} \ddot{U}(t)+\Vert\dot{U}(t)\Vert^{\alpha}\dot{U}(t)+\nabla F(U(t))=g(t),\quad t \in \mathbb{R}_+,\\ \qquad U(0)=U_{0}\,\in \mathbb{R}^{N},\quad\dot{U}(0)=U_{1}\in \mathbb{R}^{N}. \end{array}\right.$

     

Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations

We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space ${\mathbb{R}^3}$ and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniqueness for weak solutions is still open. Furthermore, we show that the solutions of the Voigt regularized system converge, as the regularization parameter ${\alpha \rightarrow 0}$ , to strong solutions of the original inviscid resistive MHD, on the corresponding time interval of existence of the latter. Moreover, we also establish a new criterion for blow-up of solutions to the original MHD system inspired by this Voigt regularization.     

On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

In (Comm Pure Appl Math 62(4):502–564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier–Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier–Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607–641, 2012) in several respects.     

Sharp Sobolev Asymptotics for Critical Anisotropic Equations

We investigate blow-up theory and prove sharp Sobolev asymptotics for a general class of anisotropic critical equations in bounded domains of .     

Regularity Results for Two-Dimensional Flows of Multiphase Viscous Fluids

Global regularity results for weak solutions of the Navier-Stokes equations for two-dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density. As an application, we deduce regularity properties for the integral curves of the corresponding velocity field. Finally, we prove regularity results “in the small” for strong solutions. (Accepted October 10, 1995)     

Regularity Results for Nonlocal Equations by Approximation

We obtain C ^1,α regularity estimates for nonlocal elliptic equations that are not necessarily translation-invariant using compactness and perturbative methods and our previous regularity results for the translation-invariant case.     

Viscosity Solutions of Dynamic-Programming Equations for the Optimal Control of the Two-Dimensional Navier-Stokes Equations

In this paper we study the infinite-dimensional Hamilton-Jacobi equation associated with the optimal feedback control of viscous hydrodynamics. We resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution.