Existence Of Strong Solutions Research Articles

Global existence and large time behavior of strong solutions to the Cauchy problem of the 3D heat conducting incompressible MHD flows

In this paper, we consider an initial value problem of inhomogeneous heat conducting magnetohydrodynamic equations in R3. Through some time-weighted a priori estimates, we prove the global existence of strong solution provided that (‖ρ0u0‖L22+‖b0‖L22)(‖∇u0‖L22+‖∇b0‖L22) is suitably small, which extends the corresponding Zhong's results [16,17] to the whole space R3. Furthermore, we also derive the algebraic decay estimates of the solution.

Neumann problem for backward SPDEs with singular terminal conditions and application in constrained stochastic control under target zone

In this paper, a Neumann problem for the backward stochastic partial differential equation (BSPDE) with singular terminal condition is studied, which characterizes the value function for a constrained stochastic control problem (also called optimal liquidation problem) in target zone models. The existence and the uniqueness of strong solutions to such BSPDEs are addressed. Furthermore, the uniqueness and existence of strong solutions to the Neumann problem for general semilinear BSPDEs in finer function spaces, a comparison theorem, and a new link between forward–backward stochastic differential equations and BSPDEs are proved as well. In addition, we apply the strong solution theory to the associated optimal liquidation problems and derive the optimal feedback control.

On the Existence of Strong Solutions to the Cahn--Hilliard--Darcy System with Mass Source

We study a diffuse interface model describing the evolution of the flow of a binary fluid in a Hele-Shaw cell. The model consists of a Cahn-Hilliard-Darcy (CHD) type system with transport and mass source. A relevant physical application is related to tumor growth dynamics, which in particular justifies the occurrence of a mass inflow. We study the initial-boundary value problem for this model and prove global existence and uniqueness of strong solutions in two space dimensions as well as local existence in three space dimensions.

Existence of Strong Solutions for Nonlinear Systems of PDEs Arising in Convective Flow

In this paper, we will study the existence of strong solutions for a nonlinear system of partial differential equations arising in convective flow, modeling a phenomenon of mixed convection created by a heated and diving plate in a porous medium saturated with a fluid. The main tools are Schäfer’s fixed-point theorem, the Fredholm alternative, and some theorems on second-order elliptic operators.

Existence of global strong solutions for a reduced gravity two-and-a-half layer model

In this paper, we consider the existence of global strong solutions to a reduced gravity two-and-a-half layer model in one-dimensional periodic domain. This work is inspired by Constantin-Drivas-Nguyen-Pasqualotto ((2020) [2]). Based on the Bresch-Desjardins entropy and the analysis of the evolution of the active potential, we obtain the continuation criterion and global existence of strong solutions when there is no vacuum. Due to the structure complexity of the reduced gravity two-and-a-half model, we still need to deal with the extra cross terms, which is different from Constantin-Drivas-Nguyen-Pasqualotto ((2020) [2]).

Local well‐posedness to the 2D Cauchy problem of nonhomogeneous heat‐conducting Navier–Stokes and magnetohydrodynamic equations with vacuum at infinity

This paper concerns the Cauchy problem of nonhomogeneous heat‐conducting magnetohydrodynamic (MHD) equations in with vacuum as far‐field density. By spatial weighted energy method, we derive the local existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. As a byproduct, we get the local existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat‐conducting Navier–Stokes equations with vacuum at infinity.

Global Well-Posedness to the 3D Cauchy Problem of Nonhomogeneous Heat Conducting Navier–Stokes Equations with Vacuum and Large Oscillations

We revisit the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. For the initial density allowing vacuum at infinity, we derive the global existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. In particular, the initial data can be arbitrarily large. This improves our previous work where the initial density has non-vacuum states at infinity. The result could also be viewed as an extension of the study in L{\"u}-Xu-Zhong for the inhomogeneous case to the full inhomogeneous situation. The method is based on delicate spatial weighted estimates and the structural characteristic of the system under consideration. As a byproduct, we get the global existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat conducting Navier-Stokes equations with vacuum at infinity.

Global existence of strong solutions for a general incompressible Oldroyd-B system without damping mechanism

In this paper, we consider the three dimensional Cauchy problem for a general incompressible Oldroyd-B system without damping mechanism, and establish the global well-posedness of strong solutions.

The global solvability of the Hall-magnetohydrodynamics system in critical Sobolev spaces

We are concerned with the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita–Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2[Formula: see text]D flows for the Hall-MHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the so-called velocity of electron [Formula: see text] and as regards [Formula: see text]D flows, of the auxiliary vector-field [Formula: see text] that comes into play in the total magneto-helicity balance for the Hall-MHD system.

The Existence of Strong Solution for Generalized Navier-Stokes Equations withpx-Power Law under Dirichlet Boundary Conditions

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized Navier-Stokes equation modeling bypx-power law with Dirichlet boundary condition under the restriction3n/n+2n+2<px<2n+1/n−1. In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction2n+1/n+3<px<2n+1/n−1, which arises from the nonflatness of domain.

Global solution to the compressible non-isothermal nematic liquid crystal equations with constant heat conductivity and vacuum

This paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat conductivity coefficients are kept to be positive constants. Global existence of strong solutions is established for any H^{2} initial data in the Lagrangian flow map coordinate, which holds for both vacuum and non-vacuum case. The key difficulty is caused by the lack of the positive lower bound of the density. To overcome such difficulty, it is observed that the ratio of frac{rho _{0(y)}}{rho (t,y)} is proportional to the time integral of the upper bound of temperature and vector director field, along the trajectory. Density weighted Sobolev type inequalities are constructed for both temperature and director field in terms of frac{rho _{0(y)}}{rho (t,y)} and small dependence on their dissipation estimates. Besides this, to deal with cross terms arising due to liquid crystal flow, higher order priori estimates are established by using effective viscous flux.

On the global existence of a generalized Shigesada-Kawasaki-Teramoto system

We establish the existence of strong solutions to a class of cross diffusion systems on RN (N≤3) which generalizes the Shigesada-Kawasaki-Teramoto (SKT) model in population dynamics. The SKT model on planar domains (N=2) with polynomial diffusions and advections is completely solved.

Global existence of strong solution to the chemotaxis-shallow water system with vacuum in a bounded domain

In this paper, we establish the global existence of strong solution to the chemotaxis-shallow water system with vacuum in a bounded domain Ω⊂R2 provided that the initial data is partially small. We first get a new blow-up criterion and then we use bootstrap argument to obtain the global existence of strong solution.

Local-in-time existence of strong solutions to a class of the compressible non-Newtonian Navier–Stokes equations

The aim of this article is to show a local-in-time existence of a strong solution to the generalized compressible Navier-Stokes equations for arbitrarily large initial data. The goal is reached by \(L^p\)-theory for linearized equations which are obtained with help of the Weis multiplier theorem and can be seen as a generalization of the work of Enomoto and Shibata (Funkcial Ekvac 56(3):441–505, 2013) (devoted to compressible fluids) to compressible non-Newtonian fluids.

$$L^p$$-strong solution to fluid-rigid body interaction system with Navier slip boundary condition

We study a fluid-structure interaction problem describing movement of a rigid body inside a bounded domain filled by a viscous fluid. The fluid is modelled by the generalized incompressible Naiver–Stokes equations which include cases of Newtonian and non-Newtonian fluids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and balance of forces at the fluid-rigid body interface. Our analysis also includes the case of the nonlinear slip condition. The main results assert the existence of strong solutions, in an $$L^p-L^q$$ setting, globally in time, for small data in the Newtonian case, while existence of strong solutions in $$L^p$$ -spaces, locally in time, is obtained for non-Newtonian case. The proof for the Newtonian fluid essentially uses the maximal regularity property of the associated linear system which is obtained by proving the $${\mathcal {R}}$$ -sectoriality of the corresponding operator. The existence and regularity result for the general non-Newtonian fluid-solid system then relies upon the previous case. Moreover, we also prove the exponential stability of the system in the Newtonian case.

Global strong solutions to the Vlasov-Poisson-Boltzmann system with soft potential in a bounded domain

Boundary effects are crucial for dynamics of dilute charged gases governed by the Vlasov-Poisson-Boltzmann (VPB) system. In this paper, we study the existence and regularity of solutions to the VPB system with soft potential in a bounded convex domain with in-flow boundary condition. We establish the existence of strong solutions in the time interval [0,T] for an arbitrary given T>0 when the initial distribution function is near an absolute Maxwellian. Our contribution is based on a new weighted energy estimate in some W1,p space and Lx3Lv1+ space for soft potential. By using the classical L2–L∞ method and bootstrap argument, we extend the local solutions from small time scale to large time scale.

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

Population cross-diffusion systems of Shigesada–Kawasaki–Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

A note on global existence of strong solution to the 3D micropolar equations with a damping term

This paper studies the Cauchy problem of the 3D incompressible micropolar equations with a damping term sigma |u|^{beta -1}u (sigma >0, 1le beta <3). It is shown that the strong solutions exist globally for any 1le beta <3.

Global well-posedness and nonlinear stability of a chemotaxis system modelling multiple sclerosis

We consider a system of reaction–diffusion equations including chemotaxis terms and coming out of the modelling of multiple sclerosis. The global existence of strong solutions to this system in any dimension is proved, and it is also shown that the solution is bounded uniformly in time. Finally, a nonlinear stability result is obtained when the chemotaxis term is not too big. We also perform numerical simulations to show the appearance of Turing patterns when the chemotaxis term is large.

Non-Newtonian two-phase thin-film problem: Local existence, uniqueness, and stability

We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the two-phase Navier–Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.