Uniqueness Of Strong Solutions Research Articles

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.

Investigations of effective dispersion models for electroosmotic flow with rigid and free boundaries in a thin strip

Reactive flow and transport in porous media is topic of intense research since decades. Since dispersion is one of the key parameters in solute transport, its accurate modeling is essential to avoid wrong predictions of flow and transport behavior. In this research, we investigate novel effective dispersion models for reactive transport of electrically charged chemical species in a thin, potentially evolving strip taking into account Taylor–Aris and electroosmotic‐induced dispersion as well as their cross‐coupling effects. We prove positivity of the dispersion coefficient and the existence and uniqueness of strong solutions in the fixed geometry setting. Moreover, we numerically investigate scenarios for both the fixed and evolving geometry situation. The simulation results illustrate the possibility of separating charged species, such that the findings of this study can lead to a better understanding of mixing and separation processes of charged solutes and an improved prediction of breakthrough curves. Finally, we study the limits of vanishing channel width, precipitation layer thickness, and molecular diffusion. We show convergence of the solutions to the corresponding limit cases such as a hyperbolic model or the fixed geometry case. From these results, we can rate the impact of distinct dispersion mechanisms and evaluate the necessity of a detailed modeling for different parameter regimes.

Theoretical and Computational Analysis of the Thermal Quasi-Geostrophic Model

This work involves theoretical and numerical analysis of the thermal quasi-geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froude number and the stratification parameter are all of the same asymptotic order. The main analytical contribution of this paper is to construct local-in-time unique strong solutions for the TQG model. For this, we show that solutions of its regularised version α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-TQG converge to solutions of TQG as its smoothing parameter α→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 0$$\\end{document} and we obtain blow-up criteria for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-TQG model. The main contribution of the computational analysis is to verify the rate of convergence of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-TQG solutions to TQG solutions as α→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 0$$\\end{document}, for example, simulations in appropriate GFD regimes.

Global well-posedness to the 3D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large oscillations and vacuum

We consider the Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with vacuum in R3. Based on delicate energy estimates, we establish the global existence and uniqueness of strong solutions under some smallness condition. Moreover, we also derive large-time decay rates of the solution. In particular, our smallness condition is independent of any norms of the initial data except the upper bound of the initial density.

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 existence and asymptotic stability of the free boundary problem of the primitive equations with heat insulation

In this paper, we deal with the free boundary value problem of the three-dimensional full primitive equations for the large-scale ocean with heat insulation in a bounded domain. The domain is periodic in the horizontal direction and finite in the vertical direction, and its upper boundary is a free surface driven by the fluid and bottom boundary is fixed. We show that the unique strong solution of the free boundary value problem for the three-dimensional full primitive equations exists globally in time and converges to the equilibrium state with exponential decay rate, when the initial data is a small perturbation of the steady state in regular Sobolev space.

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.

Error estimates of a finite volume method for the compressible Navier–Stokes–Fourier system

In this paper we study the convergence rate of a finite volume approximation of the compressible Navier–Stokes–Fourier system. To this end we first show the local existence of a regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution.

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.

Weighted mixed-norm Lp estimates for equations in non-divergence form with singular coefficients: The Dirichlet problem

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the coefficients are constants and they cover the case where the weights may not be in the Ap-Muckenhoupt class.

Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

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.

Stochastic Integral Evolution Equations with Locally Monotone and Non-Lipschitz Coefficients

In this work the existence and uniqueness of strong solutions are established for a class of stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients. In particular, our results can be applied to a large class of stochastic partial differential equation models with hereditary or memory effects such as quasilinear SPDEs like stochastic porous medium equations and semilinear SPDEs like stochastic 2D Navier—Stokes equations with hereditary or memory terms.

Local existence of strong solutions to the stochastic Navier-Stokes equations with Lp data

For the stochastic Navier-Stokes equations with a multiplicative white noise on T3, we prove that there exists a unique strong solution locally in time when the initial datum belongs to Lp(Ω;Lp) for p>3.

Analytical Properties for a Stochastic Rotating Shallow Water Model Under Location Uncertainty

The rotating shallow water model is a simplification of oceanic and atmospheric general circulation models that are used in many applications such as surge prediction, tsunami tracking and ocean modelling. In this paper we introduce a class of rotating shallow water models which are stochastically perturbed in order to incorporate model uncertainty into the underlying system. The stochasticity is chosen in a judicious way, by following the principles of location uncertainty, as introduced in Mémin (Geophys Astrophys Fluid Dyn 108(2):119–146, 2014). We prove that the resulting equation is part of a class of stochastic partial differential equations that have unique maximal strong solutions. The methodology is based on the construction of an approximating sequence of models taking value in an appropriately chosen finite-dimensional Littlewood-Paley space. Finally, we show that a distinguished element of this class of stochastic partial differential equations has a global weak solution.

On the Existence and Structures of Almost Axisymmetric Solutions to 3-D Navier–Stokes Equations

In this paper, we consider 3-D Navier–Stokes equations with almost axisymmetric initial data, which means that by writing in the cylindrical coordinates, then and are small in some sense (recall axisymmetric means these three quantities vanish). Then with additional smallness assumption on , we prove the global existence of a unique strong solution , and this solution keeps close to some axisymmetric vector field. We also establish some refined estimates for the integral average in variable for . Moreover, as and here depend on , it is natural to expand them into Fourier series in variable. And we shall consider one special form of , with some small parameter to measure its swirl part and oscillating part. We study the asymptotic expansion of the corresponding solution, and the influences between different profiles in the asymptotic expansion. In particular, we give some special symmetric structures that will persist for all time. These phenomena reflect some features of the nonlinear terms in Navier–Stokes equations.

A remark on the regularity criterion for the 3D Navier–Stokes equations in terms of two vorticity components

In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations in terms of only two vorticity components. By making use of a logarithmic Sobolev inequality in the Besov spaces with negative indices and well known commutator estimate, we prove that a unique local strong solution does not blow-up at time T if two components of vorticity belong to L2(0,T;Ḃ∞,∞−1). This result is the further extension of the previous works by Guo et al. (2018) and by O (2021), and becomes the improvement of the work by Gala (2011).

Theoretical and numerical analysis of a parabolic system with chemoattraction modeling the growth of glioma cells

In this paper, we consider a two dimensional PDE model describing the proliferation-invasion structure of hypoxic glial cells in the brain, in which the cell movement is determined not only by natural diffusion but also, by the gradient of the concentration of oxygen. This model is given by a second order nonlinear system involving the density of cancer cells and the oxygen concentration, which are coupled by a chemoattraction term, a source of logistic growth, and a reaction term of Michaelis-Menten type. The contribution of this paper is summarized in the following two main aspects: first, we prove the existence and uniqueness of strong solutions; and second, we propose a nonlinear fully discrete finite element (FE) approximation for the model. We prove its well-posedness, some uniform estimates, the positivity for the discrete oxygen concentration and approximated positivity for the discrete tumour cells, which are required in this biological model. The key point to develop the numerical analysis is to control properly the second order nonlinear chemotaxis term as well as to obtain a discrete energy estimate which, in particular, gives a bounded energy; this energy estimate is derived by using a regularization technique. Finally, we present some numerical simulations which allow us to validate the theoretical results obtained.

Strong solutions to a nonlinear stochastic aggregation-diffusion equation

It is well-known that solutions to deterministic nonlocal aggregation-diffusion models may blow up in two or higher dimensions. Various mechanisms hence have been proposed to “regularize” the deterministic aggregation-diffusion equations in a manner that allows pattern formation without blow-up. However, stochastic effect has not been ever considered among other things. In this work, we consider a nonlocal aggregation-diffusion model with multiplicative noise and establish the local existence and uniqueness of strong solutions on [Formula: see text]. If the noise is non-autonomous and linear, we establish the global existence and large-time behavior of strong solutions with decay properties by combining the Moser-Alikakos iteration technique and some decay estimates of Girsanov type processes. If the noise is nonlinear and strong enough, we show that blow-up can be prevented. As such, our results assert that certain multiplicative noise can also regularize the aggregation-diffusion model.

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.