Related papers: Global solvability for the heat equations in two h…
We examine the so-called micropolar equations in three dimensional cylindrical domains under Navier boundary conditions. These equations form a generalization of the ordinary incompressible Navier-Stokes model, taking the structure of the…
We prove a global existence theorem (with respect to a geometrically- defined time) for globally hyperbolic solutions of the vacuum Einstein equations which admit a $T^2$ isometry group with two-dimensional spacelike orbits, acting on $T^3$…
This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity $\omega_0\in W^{k,p}(\R^{2}_+)$ with $k\geq3$ and $1<p<2$, it…
This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985), which can be used as a phase transition model. We will prove the existence…
This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided…
In this paper, we study the global existence of classical solutions to the three dimensional ideal invicid compressible and heat conductive fluids with radial symmetrical data in $H^s(\mathbb{R}^3)$. Our proof is based on the symmetric…
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the…
We consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. This model was recently introduced in a previous paper of ours, where we proved existence of…
We prove small-data global existence to semi-linear wave equations on hyperbolic space of dimension greater than or equal to three, for nonlinearities that have the form of a sufficiently high integer power of the solution. We also prove…
In this paper, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system on $\mathbb T^2$, based on a new approximate system which is different from the classical Ginzburg-Landau approximation.…
We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation.…
Time-periodic weak solutions for a coupled hyperbolic-parabolic system are obtained. A linear heat and wave equation are considered on two respective $d$-dimensional spatial domains that share a common $(d-1)$-dimensional interface…
A map from the initial conditions to the values of the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with $n$ interfaces. The existence of this map…
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On…
We present a fast adaptive method for the evaluation of heat potentials, which plays a key role in the integral equation approach for the solution of the heat equation, especially in a non-stationary domain. The algorithm utilizes a…
The global well-posedness and stability of solutions to the three-dimensional compressible Euler equations with damping is a longstanding open problem. This problem was addressed in \cite{WY, STW} in the isentropic regime (i.e. $\gamma>1$)…
We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$ along…
In this paper we consider a semilinear parabolic equation with nonlinear and nonlocal boundary condition and nonnegative initial datum. We prove some global existence results. Criteria on this problem which determine whether the solutions…
In this paper, we consider the 3D primitive equations of oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation. Global…
We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up…