UNIFORM WELL-POSEDNESS FOR A TIME-DEPENDENT GINZBURG-LANDAU MODEL IN SUPERCONDUCTIVITY

Global existence of smooth solutions is proved for an effective theory of bubbly liquids for either the initial value problem or initial boundary value problem in one dimension. This shows that the theory does not describe shock waves or bubble collapse. Since the analysis is not for the steady boundary value problem, there is no discussion of resonance. The proof uses a semilinear form of the equations to get local existence. A priori bounds resulting from energy conservation and a nonlinear Gronwall‐like inequality are then derived to prove global existence.

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\|\sqrt\rho_0u_0\|_{L^2(\Omega)}^2+\|H_0\|_{L^2(\Omega)}^2$ and $\|\nabla u_0\|_{L^2(\Omega)}^2+\|\nabla H_0\|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.

A regularity criterion to the time-dependent Ginzburg-Landau model for superconductivity in [formula omitted

We consider an initial and boundary value problem for a class of possibly degenerate second-order parabolic PDEs. The condition prescribed at the lateral boundary of the domain is of non local type, requiring the time derivative of the unknown to equal a given function of the total flux through the boundary itself (such conditions arise in problems involving contact with a body where diffusivity is infinite). We give results of local-in-time existence of (weak) solutions, and then we investigate the existence of solutions defined for all positive times. We show that, due to certain nonlinearities appearing in the problem, blow up in a finite time may occur, and we give, by means of suitable a~priori estimates, criteria for the global existence of solutions, or for occurrence of blow up.

We study the initial value problem of the Davey-Stewartson systems for the elliptic-elliptic and hyperbolic-elliptic cases. The local and global existence and uniqueness of solutions in H s is shown. Also, we prove that the scattering operator carries a band in H s into H s .

Classical solutions of general ginzburg-landau equations

In this paper, we continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea ice growth. In a previous paper, the global existence and the large‐time behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here, we show that the global existence of weak solutions and the large‐time behavior are also studied under Neumann boundary condition. In this paper, we study in space dimension lower than or equal to 3.

The Landau-Lifshitz-Bloch equation is commonly used to characterize the micromagnetic phenomena near Curie temperature. In this paper, we first use the difference method to verify the local existence of smooth solutions for the Landau-Lifshitz-Bloch equations with potential functions. Then we use the energy method to verify the global existence and uniqueness of smooth solutions for periodic initial value problems as well as the Cauchy problem of the Landau-Lifshitz-Bloch equations in dimension one. Moreover, the approach adopted in this paper can be further extended to the investigation of the existence of solutions for other ferromagnetic chain equations

We are concerned with an initial boundary value problem of nonhomogeneous heat-conducting magnetohydrodynamic equations in a bounded simply connected smooth domain $$\Omega \subseteq {\mathbb {R}}^3$$ , with Navier-slip boundary conditions for the velocity and magnetic field and Neumann boundary condition for the temperature. We prove the global existence of a unique strong solution provided that $$\left( \Vert \sqrt{\rho _0}{\mathbf {u}}_0\Vert _{L^2}^2 +\Vert {\mathbf {b}}_0\Vert _{L^2}^2\right) $$ $$\left( \Vert {{\,\mathrm{curl}\,}}{\mathbf {u}}_0\Vert _{L^2}^2 +\Vert {{\,\mathrm{curl}\,}}{\mathbf {b}}_0\Vert _{L^2}^2\right) $$ is suitably small. Moreover, we also obtain large time decay rates of the solution.

In this article, we study the initial boundary value problem of the degenerate quasilinear wave equation with a strong dissipation of the form We prove global existence of solutions in Sobolev spaces and general stability estimates using multiplier method and general weighted integral inequalities. Without imposing any growth condition at the origin φ, we show that the energy of the system is bounded above by a quantity, depending on σ and φ, which tends to zero (as time goes to infinity). We also prove the optimality of decay rate of the energy for σ ≡ 1 and φ is slowly degenerates. These estimates allows us to consider large class of functions σ and φ with general growth at the origin. We give many significant examples to illustrate how to derive from our general estimates the polynomial, exponential or logarithmic decay.

We consider the questions on the decomposition of the initial boundary value problems of elasticity theory and thin bodies for some anisotropic media. In particular, the initial boundary problems of the micropolar (and classical) theory of elasticity are presented, i.e. to introduce tensor-block matrix operators (tensor-operators). In the case of an isotropic micropolar elastic medium (isotropic and transversally isotropic classical media) tensor-block matrix operators (tensor-operators) of cofactors corresponding to the tensor-block matrix operators (tensor-operators) of of the initial boundary value problems are obtained, which are splitting the initial boundary value problems. From three-dimensional decomposed initial boundary value problems the corresponding decomposed initial boundary value problems for the theory of thin bodies are obtained.

In this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity.

The well-posedness, regularity and general stability of solutions to a two-dimensional stochastic non-local delay diffusion lattice system with a time Caputo fractional operator of order [Formula: see text] are investigated in [Formula: see text] spaces for [Formula: see text]. First, the global existence and uniqueness of solutions are established by using a temporally weighted norm, the Burkholder–Davis–Gundy inequality and the Banach fixed point theorem. Then the continuous dependence of solutions on initial values is established in the sense of [Formula: see text]th moment. In particular, the [Formula: see text]th moment Hölder regularities in time and [Formula: see text]th moment general stability, including polynomial and logarithmic stability of solutions, are obtained.

Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

This paper is concerned with the strong solutions to the Cauchy problem of a simplified Ericksen-Leslie system of compressible nematic liquid crystals in two or three dimensions with vacuum as far field density. For strong solutions, some a priori decay rate (in large time) for the pressure, the spatial gradient of velocity field and the second spatial gradient of liquid crystal director field are obtained provided that the initial total energy is suitably small. Furthermore, with the help of the key decay rates, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) in two spatial dimensions.