Existence Of Strong Solutions Research Articles

Global strong solutions to the radially symmetric compressible MHD equations in 2D solid balls with arbitrary large initial data

In this paper, we prove the global existence of strong solutions to the two-dimensional compressible MHD equations with density dependent viscosity coefficients (known as Kazhikhov-Vaigant model) on 2D solid balls with arbitrary large initial smooth data where shear viscosity μ being constant and the bulk viscosity λ be a polynomial of density up to power β. The global existence of the radially symmetric strong solutions was established under Dirichlet boundary conditions for β > 1. Moreover, as long as β>max{1,γ+24}, the density is shown to be uniformly bounded with respect to time. This generalizes the previous result [Fan et al., Arch. Ration. Mech. Anal. 245, 239–278 (2022)] of the compressible Navier-Stokes equations on 2D bounded domains where they require β > 4/3 and also improves the result [Chen et al., SIAM J. Math. Anal. 54(3), 3817–3847 (2022)] of of compressible MHD equations on 2D solid balls.

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0<a<1$$ 0 < a < 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

Global existence of strong solutions to the isentropic compressible nematic liquid crystal equations in a cuboid domain

In this paper, we study the initial-boundary value problem of three-dimensional compressible nematic liquid crystal equations in a cuboid domain. We prove that the strong solution exists globally in the time provided that both the initial L1-norm of the density ‖ρ0‖L1 and the initial L3-norm of the gradient of orientation field ‖∇d0‖L3 for nematic liquid crystal flow are small enough. The main tools of proving the global well-posedness are some time-weighted a priori estimates designed for the compressible nematic liquid crystal equations.

Global well-posedness of strong solutions to the two-dimensional inhomogeneous biaxial nematic liquid crystal flow with vacuum

This paper considers the inhomogeneous biaxial nematic liquid crystal flow in a smooth bounded domain Ω⊂R2, where the velocity u and the orthogonal unit vector fields (m,n) admit the Dirichlet and Neumann boundary condition, respectively. By applying piecewise estimate and continuity method, we get the global existence of strong solutions, provided that the basic energy is suitably small. Our result may be regarded as an extension and improvement of Gong-Lin (2022) and Li-Liu-Zhong (2017) to the Neumann boundary condition, where the initial vacuum is allowed. Some new techniques are developed in order to deal with integral estimates caused by the boundary condition, and more complicated model.

Stability Analysis of Diabetes Mellitus Model in Neutrosophic Fuzzy Environment

Neutrosophic differential equation (NDE) plays a vital role in mathematical modeling on uncertainty during last few years.It is established that NDE comprises a neutrosophic number whose membership function contains three parts like as truth function, indeterministic and falsity functions.In this paper, a diabetes mellitus model has been formulated in neutrosophic fuzzy environment to get more realistic mathematical model. The initial conditions of diabetes system are considered as trapezoidal neutrosophic fuzzy number. The concept of NDE via nuetrosophic generalised derivative of type-1 and type-2 has used in this article. This is the first article in which the stability analysis of diabetes model in neutrosophic environment is introduced. The existence of strong solution and weak neutrosophic solution of the proposed system are employed in the manuscript. Also, we have exposed the comparison of the suggested model between fuzzy environment and neutrosophic environment.The all results, formulas are confirmed graphically and numerically by MATLAB software.

Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations

Abstract Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.

The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain

The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain

The global existence of strong solutions for a non-isothermal ideal gas system

The global existence of strong solutions for a non-isothermal ideal gas system

On the global stability of large Fourier mode for 3-D Navier-Stokes equations

In this paper, we first prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations with solenoidal initial data, which writes in the cylindrical coordinates is of the form: A(r,z)cos⁡Nθ+B(r,z)sin⁡Nθ, provided that N is large enough. In particular, we prove that the corresponding solution has almost the same frequency N for any positive time. The main idea of the proof is first to write the solution in trigonometrical series in θ variable and estimate the coefficients separately in some scale-invariant spaces, then we handle a sort of weighted sum of these norms of the coefficients in order to close the a priori estimate of the solution. Furthermore, we shall extend the above well-posedness result for initial data which is a linear combination of axisymmetric data without swirl and infinitely many large mode trigonometric series in the angular variable.

Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

Existence of strong solutions for a compressible fluid-solid interaction system with Navier slip boundary conditions

Existence of strong solutions for a compressible fluid-solid interaction system with Navier slip boundary conditions

Local-in-time existence of strong solutions to an averaged thick sprays model

Local-in-time existence of strong solutions to an averaged thick sprays model

Global well-posedness of strong solutions to the primitive equations without vertical diffusivity

This paper is concerned with the initial–boundary value problem of the three-dimensional primitive equations for oceanic dynamics. The global existence of strong solutions to the primitive equations without vertical diffusivity is proved under the assumption of initial data (v0,T0)∈H1.

Global Existence of Strong Solutions and Serrin-Type Blowup Criterion for 3D Combustion Model in Bounded Domains

Global Existence of Strong Solutions and Serrin-Type Blowup Criterion for 3D Combustion Model in Bounded Domains

Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models

Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models

Remarks on strong global solutions of the [formula omitted]-equation

A bounded first-order derivative with respect to x of a solution to the b-equation, which encompasses the Camassa–Holm and Degasperis–Procesi equations, serves as a crucial condition that triggers a process leading to the establishment of bounds for derivatives of the solution up to an order determined by the regularity of the initial data. Consequently, this condition ensures the global existence of strong solutions in Sobolev spaces.

On an intercritical log-modified nonlinear Schrödinger equation in two spatial dimensions

We consider a dispersive equation of Schrödinger type with a nonlinearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with Lee-Huang-Yang correction. Mathematically, it is seen to be mass supercritical and energy subcritical with a sign-indefinite nonlinearity. For the corresponding initial value problem, we prove global in-time existence of strong solutions in the energy space. Furthermore, we prove the existence and uniqueness (up to symmetries) of nonlinear ground states and the orbital stability of the set of energy minimizers. We also show that for the corresponding model in 1D a stronger stability result is available.

Optimal decay rates for the viscous two-phase model without constraints on transition to single-phase flow

In this paper, we study a free-boundary problem of the liquid-gas two-phase drift-flux model with constant viscosity, where the initial liquid and gas masses are assumed to be connected with vacuum continuously, and the pressure function takes a singular form. We utilize the flow map to reformulate the problem and prove the global existence and decay rates of strong solutions with general initial data that involve transition to pure single-phase points or regions. In particular, both the optimal decay rates of the mass functions and the decay estimates of the velocity towards its asymptotic profile as time goes to infinity are studied. As far as we are concerned, this is the first result on the global existence of strong solutions to the two-phase drift-flux model with singular pressure function and with general initial data that allow for transition to single-phase flow. It is also worth noting that our method in recovering the non-weighted estimates of the solution by utilizing the Poincaré type inequality can apply to the single-phase problem studied in Zeng (2015) [37], so as to derive the decay estimates for the velocity and relax the constraints on the initial data.

Local existence of strong solutions to micro–macro models for reactive transport in evolving porous media

AbstractTwo-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenised flow and transport equations are solved on the macroscopic scale, while effective parameters are obtained from auxiliary cell problems on possibly evolving reference geometries (micro-scale). Despite their perspective success in rendering lab/field-scale simulations computationally feasible, analytic results regarding the arising two-scale bilaterally coupled system often restrict to simplified models. In this paper, we first derive smooth dependence results concerning the partial coupling from the underlying geometry to macroscopic quantities. Therefore, alterations of the representative fluid domain are described by smooth paths of diffeomorphisms. Exploiting the gained regularity of the effective space- and time-dependent macroscopic coefficients, we present local-in-time existence results for strong solutions to the partially coupled micro–macro system using fixed-point arguments. What is more, we extend our results to the bilaterally coupled diffusive transport model including a level-set description of the evolving geometry.

Global strong solution of 3D inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity

In this paper, we investigate the Cauchy problem of the three-dimensional inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity. For the case μ(ρ)=ρ, we prove the global existence of strong solution for small initial data satisfying the compatibility condition. Furthermore, we also obtain the large time algebraic decay rates of the velocity. The innovation of this article is that it does not require a positive lower bound for viscosity.