Schrödinger-improved Boussinesq system in two space dimensions

In this article, we study the Cauchy problem for 3D fractional compressible isentropic generalized Navier-Stokes equations for viscous compressible fluid with one Levy diffusion process. We first obtain the existence and uniqueness of global strong solutions for small initial data by providing several commutators via the Littlewood-Paley theory. We then derive the $L^2$-decay rate for the highest derivative of the strong solution without decay loss by using a cancellation of a low-medium-frequency quantity. Our results improve those provided recently in [36]. For more information see https://ejde.math.txstate.edu/Volumes/2025/35/abstr.html

The initial boundary value problem for the evolution system describing geophysical flow in three‐dimensional domains was considered. The existence and uniqueness of global strong solution to the evolution system were proved under assumption on smallness of data. Moreover, solvable compatibility conditions of initial data and boundary values which guarantee the existence and uniqueness of global strong solution were discussed. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper, we consider a two-dimensional fluid-rigid body problem. The motion of the fluid is modelled by the Navier-Stokes equations, whereas the dynamics of the rigid body is governed by the conservation laws of linear and angular momentum. The rigid body is supposed to be an infinite cylinder of circular cross-section. Our main result is the existence and uniqueness of global strong solutions.

In this paper, we study the three-dimensional primitive equations for atmosphere dynamics with only horizontal diffusivity. This system is considered to have full viscosities in the momentum equations, and have horizontal diffusions in the temperature equation, specific humidity equation and liquid water content equation. First, for any H2 initial data, we use energy estimate method to establish the higher order estimations of state functions in local time. Then applying the contraction mapping principle and the Aubin–Lions lemma, we obtain the well-posedness of local strong solution. Furthermore, we use a logarithmic Sobolev inequality of the limiting type and a classic logarithmic type Gronwall inequality to obtain the higher order estimations of state functions in global time. Applying the continuity method, we prove the existence and uniqueness of global strong solution.

In this paper, the unique global strong axisymmetric solutions for the incompressible MHD equations with only vertical dissipation and vertical magnetic diffusion are established, where no smallness assumptions on the initial data are required. One of the key ingredients of the analysis is to obtain higher-order regularity estimates for velocity and magnetic fields, where the axisymmetry of the solutions and some new L 2 energy estimates play an important role.

In this paper, we consider the long-time dynamics for the primitive equations of large-scale dry atmosphere. First, by energy methods, we obtain the existence and uniqueness of global strong solutions of the problem. Second, by studying the long-time behavior of strong solutions, we construct a global attractor which captures all the trajectories.

The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.

In this paper, we investigate the 3D magneto-micropolar Bénard equations with nonlinear damping and partial viscosity. By employing energy methods and properties in anisotropic Lebesgue spaces, we establish the existence and uniqueness of global strong solutions when $ \beta \geq 4 $. Our results not only extend the current theoretical framework but also provide new perspectives and methodologies for further research in the dynamics of magneto-micropolar fluids.

A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system (4) has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system (3) from Lρ to L2ρ (ρ≥1) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N≤3. Lastly, by using various analytical methods, a threshold value ξ00(ξ00<0) is found, such that positive constant steady state (u∗,v∗,w∗) becomes unstable when ξ<ξ00. Our results show that the haptotaxis plays a crucial role in determining the stability to the model (3), that is, it can have a destabilizing effect.

We study the three‐dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. First, we study the existence, uniqueness, and regularity of global weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since the system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and their properties.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $\gamma\ge1$. Initial data and solutions are small in $L^2$ around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when $\gamma=1$. A great deal of information about partial regularity and large-time behavior is obtained.

On some nonlinear hyperbolic systems with damping boundary conditions

A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations

In this paper we study the global existence and asymptotic behavior of solutions for the Maxwell-Schrodinger equations under the Coulomb gauge condition in three space dimensions with the final states given att=+∞. This leads to the construction of the modified wave operator for certain scattered data. It is also shown that for the initial data in the range of the modified wave operator, the initial value problem of the Maxwell-Schrodinger equations has the global solutions in time.