Unique Global Strong Solution Research Articles

Global well-posedness for 2D non-resistive MHD equations in half-space

This paper focuses on the initial boundary value problem of two-dimensional non-resistive MHD equations in a half space. We prove that the non-resistive MHD equations have a unique global strong solution around the equilibrium state (0, e1) for Dirichlet boundary condition of velocity and modified Neumann boundary condition of magnetic.

$$C^{\infty }$$-Regularization by Noise of Singular ODE’s

AbstractIn this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the “roughness” and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds, in a certain sense, to the Nash–Moser iterative scheme in combination with a new concept of “higher order averaging operators along highly fractal stochastic curves”. This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

Global well-posedness to the 3D incompressible magneto-micropolar Bénard system with damping and zero thermal conductivity

We are concerned with the three-dimensional (3D) Cauchy problem of the incompressible magneto-micropolar Bénard system with a nonlinear damping term α|u|β−1u(α>0andβ≥1) in the momentum equations. By energy method, we derive a unique global strong solution for such a model when β≥4. Our result extends previous related works.

A hyperbolic-elliptic-parabolic PDE model of chemotactic E. coli colonies

This paper studies the hyperbolic-elliptic-parabolic system{ut=−∇⋅(u∇c)+g(u)nu−b(n)u1+θ,(x,t)∈Ω×R+,0=Δc+αu−βc,(x,t)∈Ω×R+,nt=Δn−γg(u)nu,(x,t)∈Ω×R+,∂c∂ν=∂n∂ν=0,(x,t)∈∂Ω×R+,u(x,0)=u0(x),n(x,0)=n0(x),x∈Ω in a bounded domain Ω⊂RN(N≥1) with smooth boundary, where g(s) and b(s) are nonnegative smooth functions, α,β,γ>0 and θ∈[0,∞). We prove that if θ>1 or θ=1 with lims→+∞⁡b(s)=b0>α, the system admits a unique global strong solution, while if θ∈[0,1), the system exhibits a finite-time explosion phenomenon in the sense of L∞ norm for some large ‖u0‖Lp(Ω) with p>N. Here the convexity of the domain Ω is not required.

Global strong solutions to the one-dimensional compressible Navier-Stokes-Allen-Cahn system with phase field variable dependent viscosity

In this paper, we study the global existence and large-time behavior of strong solutions to the Cauchy problem of the one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system with phase field variable dependent viscosity, which models the motion of a mixture of two viscous compressible fluids. The case when the pressure p(ρ)=ργ, the viscosity ν(ρ,χ)=χα, the interface thickness δ(ρ)=ρβ and the relaxation time function a(ρ,χ,χy)=χλ is considered, where ρ>0 and χ are the density and the phase field variable, respectively, and γ,α,β,λ∈R are parameters. Under some conditions on the parameters γ,α,β,λ and the initial data, it is shown that the Cauchy problem of the one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system admits a unique global strong nonvacuum solution, which tends to the constant equilibrium state as time goes to infinity. Moreover, the exponential time decay rate of the phase field variable χ is also obtained. Here the initial perturbation for the phase field variable χ should be small, but the initial perturbations for the density and velocity of the fluid can be arbitrarily large.

Global regularity and asymptotic stabilization for the incompressible Navier–Stokes-Cahn–Hilliard model with unmatched densities

We study an initial-boundary value problem for the incompressible Navier–Stokes–Cahn–Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\rightarrow \\infty $$\\end{document}. More precisely, the concentration function ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} is a strong solution of the Cahn–Hilliard equation for (arbitrary) positive times, whereas the velocity field u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{{u}}}$$\\end{document} becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn–Hilliard equation with divergence-free velocity belonging only to L2(0,∞;H0,σ1(Ω))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(0,\\infty ; {\ extbf{H}}^1_{0,\\sigma }(\\Omega ))$$\\end{document}, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz–Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.

Global well‐posedness for the one‐phase Muskat problem

AbstractThe free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.

Global strong solutions and large time behavior to a micro-macro model for compressible polymeric fluids near equilibrium

In this paper, we mainly study the global strong solutions to a micro-macro model for compressible polymeric fluids with small initial data and their long-time decay rates of all order spatial derivatives. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first present that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for d≥2. Moreover, for d≥3, we show a new critical Fourier estimation that allows us to derive the long-time decay rates of the L2 norm for all order spatial derivatives.

An optimal control problem for a biological population model with diffusion and infectious disease

This paper studies the optimal control problem for a biological population model with diffusion and infectious diseases, and we intend to maximize the density of species with proper control of predator populations. First, we prove that the controlled system admits a unique global positive strong solution for any given admissible control variable. Then, we show that the optimal control problem admits at least one optimal pair by applying a compactness argument. Moreover, by introducing appropriate dual variables for the linearized control system, we establish bang-bang-type first-order necessary optimality conditions for the optimal pair. Second-order necessary and sufficient conditions for the optimal pair are also provided. Finally, a concrete numerical example is given to verify the results of this paper.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

Full compressible Navier-Stokes equations with the Robin boundary condition on temperature

We consider full compressible Navier-Stokes equations with the Robin boundary condition on temperature. Note that the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature. It is shown that a unique global strong solution existed if the initial data belongs to H 1 . Subsequently, we find that the strong solution is nonlinearly exponentially stable as time tends to infinity. This result could be viewed as the first one on the global well-posedness of the strong solution to full Navier-Stokes equations in a bounded domain with the degenerate heat conductivity and the Robin boundary condition on temperature. The proofs are mainly based on the energy method and a special inequality.

Strong solutions for the Cauchy problem to three-dimensional nonhomogeneous incompressible micropolar fluid equations with vacuum

This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible micropolar fluid equations in the whole space. We first establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous micropolar equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the micro-rotational velocity. Then as an immediate application, we prove that the Cauchy problem of micropolar fluid equations has a unique global strong solution, provided that the kinematic viscosity is sufficiently large, or the upper bound of initial density or initial kinetic energy is small enough.

Regularity results in 2D fluid–structure interaction

We study the interaction of an incompressible fluid in two dimensions with an elastic structure yielding the moving boundary of the physical domain. The displacement of the structure is described by a linear viscoelastic beam equation. Our main result is the existence of a unique global strong solution. Previously, only the ideal case of a flat reference geometry was considered such that the structure can only move in vertical direction. We allow for a general geometric set-up, where the structure can even occupy the complete boundary. Our main tool—being of independent interest—is a maximal regularity estimate for the steady Stokes system in domains with minimal boundary regularity. In particular, we can control the velocity field in W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} in terms of a forcing in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} provided the boundary belongs roughly to W3/2,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{3/2,2}.$$\\end{document} This is applied to the momentum equation in the moving domain (for a fixed time) with the material derivative as right-hand side. Since the moving boundary belongs a priori only to the class W2,2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2},$$\\end{document} known results do not apply here as they require a C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document}-boundary.

Global well-posedeness and large time asymptotic behavior of 2D density-dependent Boussinesq equations of Korteweg type with vacuum

In this paper, we are devoted to study the Cauchy problem of the incompressible density-dependent Boussinesq equations of Korteweg type with vacuum. We establish some key spatial weighted estimates and a priori decay-in-time rate of the strong solutions by using the energy method. Furthermore, we prove that there is a unique global strong solution for the 2D Cauchy problem when the spatial weighted norm of the initial density is suitably small. Finally, we also obtain the large time decay rates for the gradients of velocity, temperature and pressure.

Global well-posedness for the compressible non-resistive MHD equations in a 3D infinite slab

In this paper, we investigate the existence and uniqueness of global strong solutions to the system of compressible viscous non-resistive MHD equations in a three-dimensional horizontally infinite slab with finite height. We first reformulate our analysis to Lagrangian coordinates, and then develop a new mathematical approach to establish global well-posedness of the MHD system, which requires no compatibility conditions on the initial data.

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

On the 3D Incompressible Boussinesq Equations in a Class of Variant Spherical Coordinates

This paper investigates the global stabilizing effects of the geometry of the domain at which the flow locates and of the geometric structure of the solution to the incompressible flows by studying the three-dimensional (3D) incompressible, viscosity, and diffusivity Boussinesq system in spherical coordinates. We establish the global existence and uniqueness of the smooth solution to the Cauchy problem for a full 3D incompressible Boussinesq system in a class of variant spherical coordinates for a class of smooth large initial data. We also construct one class of nonempty bounded domains in the three-dimensional space ℝ 3 , in which the initial boundary value problem for the full 3D Boussinesq system in a class of variant spherical coordinates with a class of large smooth initial data with swirl has a unique global strong or smooth solution with exponential decay rate in time.

Global existence of strong solutions to the multi-dimensional inhomogeneous incompressible MHD equations

This paper is concerned with the Cauchy problem of the multi-dimensional incompressible magnetohydrodynamic equations with inhomogeneous density and fractional dissipation. It is shown that when α+β=1+n2 satisfying 1≤β≤α≤min{3β2,n2,1+n4} and max{n4,n+16}<α for n≥3, the inhomogeneous incompressible MHD equations have a unique global strong solution for the initial data in some Sobolev spaces without requiring small conditions.

Global well-posedness for axisymmetric MHD equations with vertical dissipation and vertical magnetic diffusion

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.

Schrödinger-improved Boussinesq system in two space dimensions

We study the Cauchy problem for the Schrödinger-improved Boussinesq system in a two-dimensional domain. Under natural assumptions on the data without smallness, we prove the existence and uniqueness of global strong solutions. Moreover, we consider the vanishing “improvement” limit of global solutions as the coefficient of the linear term of the highest order in the equation of ion sound waves tends to zero. Under the same smallness assumption on the data as in the Zakharov case, solutions in the vanishing “improvement” limit are shown to satisfy the Zakharov system.