Global Existence Of Smooth Solutions Research Articles

Global smooth solutions of multidimensional compressible Euler equations with critical time depending damping and vorticity

The authors (Hou and Yin, 2017 [9]; Hou et al., 2018 [10]) proved that the 2D and 3D irrotational compressible Euler equations with critical time depending damping admit global classical solutions. In this paper, we will study this problem with non-zero vorticity and show the global existence of smooth solutions to the 2D and 3D compressible Euler equations with critical time depending damping. They used the Lorentz vector field and their method fails in the presence of the vorticity. The main ingredients in this paper are the auxiliary energies which will make up for the loss of the Lorentz vector field and the improved decay estimates of the energy which can be achieved by a new multiplier.

Long time asymptotics of small mass solutions for a chemotaxis-consumption system involving prescribed signal concentrations on the boundary

This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity χ=χ(c) under no-flux/Dirichlet boundary conditions. For general χ which may allow singularities at c=0, the global existence and boundedness of radial large data solutions are established in dimensions d≥2. In particular, when χ(c)=1, we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with χ(c)=c, a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial (d=2) and radial (d≥2) solutions at suitably small mass levels.

Existence of a global smooth solution for a degenerate problem of the hyperbolic conservation law systems

In this paper, we consider the existence of a global smooth solution for a degenerate problem of the hyperbolic conservation law systems. We use the characteristic decomposition to establish the global existence of smooth solutions to the degenerate Goursat problem up to a boundary and prove that the vacuum bubble is a point. This problem comes from the expansion of a wedge of gas with constant velocity into vacuum in gas dynamics.

Hard congestion limit of the dissipative Aw–Rascle system

In this study, we analyse the famous Aw–Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called ‘hard congestion limit’, and show the convergence of a subsequence of solutions towards a weak solution of a hybrid free-congested system. This is also illustrated numerically using a numerical scheme proposed for the model studied. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contacts between the grains.

Global well-posedness and decay estimates for the one-dimensional models of blood flow with a general parabolic velocity profile

In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier–Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in Lp (2≤p≤∞) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system.

Global existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert flow with Dzyaloshinskii–Moriya interaction in two‐dimensional torus and ℝ2

This paper focuses on establishing the existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert equation with Dzyaloshinskii–Moriya (DM) interaction and V flow in two‐dimensional domains, including the torus and Euclidean space. The primary objective is to establish global existence results by investigating the conditions under which a smooth solution exists for all time, provided that the norm of the initial magnetization gradient is sufficiently small. Rigorous mathematical proofs for these global existence results are provided by combining analytical techniques and energy estimates. These findings enhance our understanding of solution behavior and regularity in the Landau–Lifshitz–Gilbert equation with DM interaction, shedding light on its dynamics in different spatial domains.

Global solutions to the bipolar non-isentropic Euler-Maxwell system in the Besov framework

ABSTRACT This paper investigates the bipolar non-isentropic compressible Euler-Maxwell system in R 3 and T 3 . For both problems, we establish the global existence of smooth solutions in the general Besov spaces, which covers the usual Sobolev spaces with higher regularity and the critical Besov space, when the initial perturbations around the constant states are small enough. As a byproduct, we obtain the large-time asymptotic behavior of the global solutions near the equilibrium state in the general Besov spaces with relatively lower regularity. The proof is based on the technical Fourier frequency-localization method developed through the Littlewood-Paley theory, but some new development and technique are proposed for treating the strong coupling and nonlinearity for the bipolar non-isentropic case.

Global Existence of Smooth Solutions for the Diffusion Approximation Model of General Gas in Radiation Hydrodynamics

Global Existence of Smooth Solutions for the Diffusion Approximation Model of General Gas in Radiation Hydrodynamics

Global existence and decay estimate of solution to rate type viscoelastic fluids

We are concerned with the global well-posedness of rate type viscoelastic fluids system (1.1)–(1.2). First, we prove the global existence and decay estimate of solution to (1.1)–(1.2) with stress diffusion near a constant state in HN(R3) (N≥4). We only assume that the H3-norm of initial data is small, but the L2-norm of the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev or Besov spaces, we obtain the optimal decay rates of the solution and its higher order derivatives. Next, we establish the local existence of smooth solution to (1.1)–(1.2) without stress diffusion and present a blow-up criterion in R2. From which and the Bony decomposition, we prove the global existence of smooth solution.

Existence of Radial Global Smooth Solutions to the Pressureless Euler–Poisson Equations with Quadratic Confinement

We consider the pressureless Euler–Poisson equations with quadratic confinement. For spatial dimension dge 2,,dne 4, we give a necessary and sufficient condition for the existence of radial global smooth solutions, which is formulated explicitly in terms of the initial data. This condition appears to be much more restrictive than the critical-threshold conditions commonly seen in the study of Euler-type equations. To obtain our results, the key observation is that every characteristic satisfies a periodic ODE system, and the existence of a global smooth solution requires the period of every characteristic to be identical.

On global smooth small data solutions of 3-D quasilinear Klein-Gordon equations on $ \mathbb{R}^{2}\times \mathbb{T} $

In this paper, for the small initial data in the suitably weighted Sobolev spaces on the product space $ \mathbb{R}^2\times\mathbb{T} $, we establish the global existence of smooth solutions to the 3-D quasilinear Klein-Gordon equations with quadratic nonlinearities. It is noted that the topic on the well-posedness of the nonlinear hyperbolic equations on product spaces arises from the studies for the propagation of waves along infinite homogeneous waveguides and the Kaluza-Klein theory. Our main result is based on the method of normal form including the parameters $ n\in\Bbb Z $ and the continuous induction method. In addition, the free profile of the solution is obtained by applying the weighted energy estimates and the temporal decay estimates.

Global existence of smooth solutions to a full Euler–Poisson system in one space dimension

When a strictly dissipative term arises in the energy equation, the Cauchy problem for Euler–Poisson systems admits global smooth solutions for small initial data. This was proved in previous studies. In this paper, we study a stability problem for a full Euler–Poisson system without dissipation in the energy equation. We prove the global existence of smooth solutions near constant equilibrium states in one space dimension. The stability results are obtained for both one-fluid and two-fluid Euler–Poisson systems. In our proof of these results, we show an L2 energy equality in Euler coordinates. Then, we establish energy estimates of derivatives of the solution in Lagrangian coordinates by a characteristic technique. These estimates together with the equivalence of smooth solutions between the two coordinates yield the global existence of the solution.

Global solutions to an initial boundary problem for the compressible 3D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains

An initial boundary value problem for compressible magnetohydrodynamics (MHD) is considered on an exterior domain (with the vanishing first Betti number) in in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier–Stokes equations with Navier-slip boundary conditions on exterior domains in three dimensions, which was not available in literature prior to the work in this paper, to the best of knowledge of the authors’.

Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas

Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas

Global well-posedness for 2D non-resistive compressible MHD system in periodic domain

This paper focuses on the 2D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion in a periodic domain. We present a systematic approach to establishing the global existence of smooth solutions when the initial data is close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. When there is no magnetic diffusion, the magnetic field and the density are governed by forced transport equations and the problem considered here is difficult. This paper implements several key observations and ideas to maximize the enhanced dissipation due to hidden structures and interactions. In particular, the weak smoothing and stabilization generated by the background magnetic field and the extra regularization in the divergence part of the velocity field are fully exploited. Compared with the previous works, this paper appears to be the first to investigate such system on bounded domains and the first to solve this problem by pure energy estimates, which help reduce the complexity in other approaches. In addition, this paper combines the well-posedness with the precise large-time behavior, a strategy that can be extended to higher dimensions.

On global smooth solutions of the 3D spherically symmetric Euler equations with time-dependent damping and physical vacuum

In this paper, we consider the global existence and convergence of smooth solutions for the three dimensional spherically symmetric compressible Euler equations with time-dependent damping and physical vacuum. The damping coefficient decays with time and the sound speed is C 1/2-Hölder continuous across the physical vacuum boundary. Both the degeneration of the damping coefficient at time infinity and the non C 1 continuity of the sound speed across the vacuum boundary will cause difficulty in proving the global existence of smooth solutions. Under suitable assumptions on the decayed damping coefficients, the globally in-time smooth solutions and convergence to the modified Barenblatt solution will be given. Also obtained are the pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary. Our result extends that in Zeng (2017 Arch. Ration. Mech. Anal. 226 33–82) by considering the degenerate damping coefficient instead of the constant damping coefficient and that in Pan (2021 Calc. Var. Partial Differ. Equ. 60 5) from the one dimensional case to the three dimensional case with spherically symmetric data.

Global quasi-neutral limit for a two-fluid Euler-Poisson system in one space dimension

The quasi-neutral limit of one-fluid Euler-Poisson systems leads to incompressible Euler equations. It was widely studied in previous works. In this paper, we deal with the quasi-neutral limit in a two-fluid Euler-Poisson system. This limit presents a different feature since it leads to compressible Euler equations. We justify this limit for global smooth solutions near constant equilibrium states in one space dimension. Specifically, we prove a global existence of smooth solutions by establishing uniform energy estimates with respect to the Debye length and the time. This allows to pass to the limit in the system for all time. Moreover, we establish global error estimates between the solution of the two-fluid Euler-Poisson system and that of the compressible Euler equations. The proof is based on classical uniform energy estimates together with various dissipation estimates. In order to control the quasi-neutrality of the velocities of two-fluids, similar conditions on the initial data are needed.

Relativistic Kinetic Theory of Polyatomic Gases: Classical Limit of a New Hierarchy of Moments and Qualitative Analysis

A relativistic version of the Kinetic Theory for polyatomic gas is considered and a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal modes is presented. In the first part, we prove via classical limit that the truncated system of moments dictates a precise hierarchy of moments in the classical framework. In the second part, we consider the particular physical case of fifteen moments closed via maximum entropy principle in a neighborhood of equilibrium state. We prove that this symmetric hyperbolic system satisfies all the general assumptions of some theorems that guarantee the global existence of smooth solutions for initial data sufficiently small.

Global Convergence to Compressible Full Navier–Stokes Equations by Approximation with Oldroyd-Type Constitutive Laws

We consider smooth solutions to a relaxed Euler system with Oldroyd-type constitutive laws. This system is derived from the one-dimensional compressible full Navier-Stokes equations for a Newtonian fluid by using the Cattaneo–Christov model and the Oldroyd-B model. In a neighborhood of equilibrium states, we construct an explicit symmetrizer and show that the system is symmetrizable hyperbolic with partial dissipation. Moreover, by establishing uniform estimates with respect to the relaxation times, we prove the uniform global existence of smooth solutions and the global-in-time convergence of the system towards the full Navier–Stokes equations.

The global solvability of the Cauchy problem for a multi-dimensional chemotaxis-Navier–Stokes system modeling coral fertilization

This paper is dedicated to the analysis of the Cauchy problem for a chemotaxis-Navier–Stokes system modeling coral fertilization in spatial dimensions two and three. We first present the unique local solvability of a smooth solution to the system for large initial data and then establish some blow-up criteria of the solution. In the whole plane, the global existence of smooth solutions to the model for a large class of initial data is constructed. Finally, we also prove the global existence of smooth solutions under the some explicit smallness conditions of initial data. In particular, we present the time decay rates of the solution in L∞ by using the De Giorgi method.