Global Existence Result Research Articles

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void

Abstract This article is devoted to the global existence and extinction behavior of the weak solution to a fast diffusion p p -Laplace equation with logarithmic nonlinearity and special medium void. By applying energy estimates approach, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, the global existence result is proved and the sufficient conditions on the occurrence of the extinction and nonextinction phenomena are given.

On a quenching problem of hyperbolic MEMS equation with a nonlocal term

We consider a nonlocal hyperbolic problem. It arises in the field of MEMS control. First, we review the stationary problem and state that there exists a unique and no solution for small and large parameter, respectively. Next, we introduce the results of global existence and dynamical properties of solution for the hyperbolic equation. Finally, we derive the result of the quenching behaviour. As proven in similar problems, we show that the quenching occurs for large parameter. This criterion is also expressed by means of the first eigenvalue of Laplace operator.

A conservative stochastic Dirac-Klein-Gordon system

Considered herein is a particular nonlinear dispersive stochastic system consisting of Dirac and Klein-Gordon equations. They are coupled by nonlinear terms due to the Yukawa interaction. We consider a case of homogeneous multiplicative noise that seems to be very natural from the perspective of the least action formalism. We are able to show existence and uniqueness of a corresponding Cauchy problem in Bourgain spaces. Moreover, the regarded model implies charge conservation, known for the deterministic analogue of the system, and this is used to prove a global existence result for suitable initial data.

Global existence and stability of the wave equation with boundary variable damping

AbstractIn this paper, we present the result of global existence of solution for the wave equation with boundary variable damping term. Then, we prove that this global solution is stable. Our study is based on the semi‐groups theory and some integral inequalities.

Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant \[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \] in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$ with smooth boundary $\partial \Omega$ . It is shown that if $m>\max \{1,\,\frac {3N-2}{2N+2}\}$ , for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$ in an appropriate sense as $t\rightarrow \infty$ , where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$ . This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys.58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys.66 (2015), 3159–3179) to $m>\frac {3N-2}{2N}$ in the case $N\geq 3$ , but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B22 (2017), 669–686) to $m>\frac {3N}{2N+2}$ when $N\geq 2$ .

Convergent scheme for a non-local transport system modeling dislocations dynamics

In this paper, we consider a diagonal hyperbolic, not necessarily strictly hyperbolic, nonlinear and non-local system of transport equations that arises in the theory of dislocations densities dynamics. Under certain monotony assumptions on the initial data, global existence and uniqueness results of Hloc1 solutions for this system was proved in Hajj (2007). We propose a natural implicit scheme verifying a gradient L2 estimate at the discrete level compatible with that obtained in the continuous case in Hajj (2007). Which leads us to prove the convergence of the numerical solution to the continuous one. To our knowledge, this is the first convergent scheme proposed for this model.

On the stability of stationary compressible Navier–Stokes flows in 3D

AbstractThis paper studies the stability of the stationary solution of the compressible Navier–Stokes equation in the 3D whole space with an external force which decays at spatial infinity. We obtain the global existence result of the non-stationary problem under the smallness assumptions on the initial perturbation around the small stationary solution. We also derive the time decay rates of the perturbations under the smallness assumption on the initial perturbations, and show the optimality of the time decay rates. The proofs are based on the combination of the spectral analysis and energy method in the framework of Besov spaces. The time-space integral estimate for the linearized semigroup around the constant state in some Besov spaces plays a crucial role in the proofs.

Global solutions of two‐dimensional inviscid and resistive magnetohydrodynamics system near an equilibrium

We investigate the global existence of classical solutions for two‐dimensional incompressible, inviscid, and resistive magnetohydrodynamics (MHD) system. Assuming that the initial magnetic field is close to an equilibrium state of constant magnetic vorticity, the global existence result could be obtained using a time‐weighted energy estimate.

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 Regularity for Gravity Unstable Muskat Bubbles

In this paper, we study the dynamics of fluids in porous media governed by Darcy’s law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.

Initial boundary value problem of pseudo‐parabolic Kirchhoff equations with logarithmic nonlinearity

In this paper, we consider the initial boundary value problem for a pseudo‐parabolic Kirchhoff equation with logarithmic nonlinearity. Using the potential well method, we obtain a threshold result of global existence and finite‐time blow‐up for the weak solutions with initial energy . When the initial energy , we find another criterion for the vanishing solution and blow‐up solution. We also establish the decay rate of the global solution and estimate the life span of the blow‐up solution. Meanwhile, we study the existence of the ground state solution to the corresponding stationary problem.

Classification of Blow-up and Global Existence of Solutions to a System of Petrovsky Equations

In this paper, we investigate global existence, uniform decay, and blow-up of solutions for a class of system of Petrovsky equations containing nonlinear damping and sources. By introducing a family of potential wells, we not only obtain the invariant sets and vacuum isolating of solutions but also give some threshold results of global existence and nonexistence of solutions. Furthermore, by using energy techniques, we also establish certain qualitative estimates for solution.

Asymptotic behaviour of the Einstein-Yang-Mills-Higgs system in a Bianchi type I model

We study the Einstein-Yang-Mills-Higgs (EYMH) system with a positive cosmological constant in the Bianchi type I space-time with locally rotational symmetry (LRS). In particular, we consider the nonlinear interaction of the Higgs field with the Yang-Mills field coupled to an unknown gravitational field. For the considered model, from certain additional conditions (the temporal gauge and some symmetries), we derive the conservation laws for the field equations and we then deduce the exact formulation of equations in the geometric framework. Furthermore, using an iterative approch and some mathematical analysis tools, we study the above system of equations. We then establish a global existence result for the homogeneous solution and we analyse its asymptotic behaviour.

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ≥0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.

Existence of standing and traveling waves in quantum hydrodynamics with viscosity

In this paper, we consider two quantum hydrodynamics systems with linear and nonlinear viscosity. For the former, we prove existence of standing waves; moreover, global existence of traveling waves is proved without restrictions on the viscosity and dispersion parameters, thanks to a suitable Lyapunov function. This is an improvement with respect to the global existence result by Lattanzio, Marcati, and Zhelyazov [Phys. D 402 (2020), 132222], where it was required that the viscosity is sufficiently strong. In addition, we discuss existence of standing waves for the latter system.

A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model

In the first part of this article, we present a new proof for Korn’s inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude.

Properties of Weak Solutions for a Pseudoparabolic Equation with Logarithmic Nonlinearity of Variable Exponents

In this paper, a new pseudoparabolic equation with logarithmic nonlinearity of variable exponents is investigated. By using the energy functional and the classical potential well, we obtain the global existence and blow-up results of weak solutions with variable exponents. These results extend and improve some recent results in which the blow-up results were showed involving the logarithmic nonlinearity with variable exponents.

Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime

We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent p satisfies 2≤p<1+2, we establish the sharp upper bound of lifespan estimate, while for the most delicate critical power p=1+2, we show that the lifespan satisfiesT(ε)≤exp⁡(Cε−(2+2)), the optimality of which remains to be proved. The key novelty is that the compact support of the initial data can be close to the event horizon. By combining the global existence result for p>1+2 obtained by Lindblad et al. (2014) [25], we then give a positive answer to the interesting question posed by Dafermos and Rodnianski (2005) [22, the end of the first paragraph in page 1151]: p=1+2 is exactly the critical power of p separating stability and blow-up.

A model of a nonisothermal two‐phase flow of compressible fluids

We introduce a simple model of the time evolution of a binary mixture of compressible fluids including the thermal effects. Despite its apparent simplicity, the model is thermodynamically consistent admitting an entropy balance equation. We introduce a suitable weak formulation of the problem based on a combination of the entropy inequality with the total energy conservation principle. Finally, we show compactness of any bounded family of weak solutions and establish a global existence result.