On the global existence of smooth solutions to the multi-dimensional compressible euler equations with time-depending damping in half space

This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term -μ/((1+t)λ)ρu, where λ≥ 0 and μ 0 are constants. We have showed that, for all λ≥ 0 and μ 0, the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initialboundary value problem in the half space R_+~d with space dimension d = 2, 3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤λ 1when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.     

Existence of global weak solutions for Navier-Stokes equations with large flux

Global existence of weak solutions to the Navier-Stokes equations in a cylindrical domain under boundary slip conditions and with inflow and outflow is proved. To prove the energy estimate, crucial for the proof, we use the Hopf function. This makes it possible to derive an estimate such that the inflow and outflow need not vanish as t→∞. The proof requires estimates in weighted Sobolev spaces for solutions to the Poisson equation. Our result is the first step towards proving the existence of global regular special solutions to the Navier-Stokes equations with inflow and outflow.     

Existence and uniqueness of global solutions for wave equations with scalar nonlinearities

We prove an existence and uniqueness theorem of global solutions for wave equations with scalar nonlinearities. Our paper is a generalization of the work of D. Kremer [4].

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Zusammenfassung Es wird ein Existenz- und Eindeutigkeitssatz für globale Lösungen von Wellengleichungen mit skalaren Nichtlinearitäten bewiesen. Die Arbeit stellt eine Verallgemeinerung der Arbeit von D. Kremer [4] dar.

     

Existence of global solutions to multidimensional equations for Bingham fluids

We consider equations describing the multidimensional motion of compressible viscous (non-Newtonian) Bingham-type fluids, i.e., fluids with multivalued function relating the stresses to the tensor of strain rates. We prove the global existence theorem in time and in the initial data for the first initial boundary-value problem corresponding to flows in a bounded domain in the class of “weak” generalized solutions. In this case, we admit an anisotropic relation between the stress and strain rate tensors and study admissible relations of this kind in detail.     

Existence and non-existence of global solutions for uniformly parabolic equations

In this paper, we prove the existence of Fujita-type critical exponents for x-dependent fully nonlinear uniformly parabolic equations of the type $$(*)\quad \partial_{t}u=F(D^{2}u,x)+u^{p}\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ These exponents, which we denote by p(F), determine two intervals for the p values: in ]1,p(F)[, the positive solutions have finite-time blow-up, and in ]p(F), +∞[, global solutions exist. The exponent p(F)?=?1?+?1/α(F) is characterized by the long-time behavior of the solutions of the equation without reaction terms $$\partial_{t}u=F(D^{2}u,x)\quad{\rm in}\ \ \mathbb{R}^{N}\times\mathbb{R}^{+}.$$ When F is a x-independent operator and p is the critical exponent, that is, p?=?p(F). We prove as main result of this paper that any non-negative solution to (*) has finite-time blow-up. With this more delicate critical situation together with the results of Meneses and Quaas (J Math Anal Appl 376:514–527, 2011), we completely extend the classical result for the semi-linear problem.     

Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations

Summary Conditions are given for the nonlinear differential equation (1)L _n y+f(t, y, ..., ...,y ^(n–1)=0to have solutions which exist on a given interval [t₀, )and behave in some sense like specified solutions of the linear equation (2)L _n z=0as t.The global nature of these results is unusual as compared to most theorems of this kind, which guarantee the existence of solutions of (1)only for sufficiently large t. The main theorem requires no assumptions regarding oscillation or nonoscillation of solutions of (2).A second theorem is specifically applicable to the situation where (2)is disconjugate on [t ₀, ),and a corollary of the latter applies to the case where Lz=z ⁿ.     

Existence of global solutions of some ordinary differential equations

Existence of globally defined solutions of ordinary differential equations is considered. The article studies the situation when most of the solutions run away to infinity in a finite time interval, but between them there exists at least one solution which is defined at all times.     

Existence of global smooth solution to the relativistic Euler equations

In this paper, we prove an existence theorem of global smooth solutions to the Cauchy problem for the one-dimensional relativistic Euler equations. The analysis is based on a priori estimates which are obtained by the characteristic method and maximum principle.     

Existence of strong solutions and global attractors for the suspension bridge equations

In this paper, we show the existence of strong solutions for the suspension bridge equations. Furthermore, the existence of strong global attractors is investigated using a new semigroup scheme.     

Existence of global solutions for second order impulsive abstract partial differential equations

In this paper, we study the existence of global solutions for a class of second order impulsive abstract functional differential equations. The results are obtained by using Leray-Schauder’s Alternative fixed point theorem. An application is provided to illustrate the theory.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations

For a general class of autonomous quasi-linear elliptic equations on we prove the existence of a least energy solution and show that all least energy solutions do not change sign and are radially symmetric up to a translation in .     

Existence of global solutions to isentropic gas dynamics equations with a source term

In this paper we prove existence of isentropic gas dynamic equations with a source term (1.2). To this end we construct a sequence of regular hyperbolic systems (1.1) to approximate the inhomogeneous system of isentropic gas dynamics (1.2). First,for each fixed approximation parameter δ and very general condition on P (ρ),we establish the existence of entropy solutions for the Cauchy problem (1.1) with bounded initial date (1.4). Second,letting=o(δ),we obtain a complete proof of the H-1loc compactness of weak entropy pairs of system (1.2) in the form η(ρ,u) =ρH(ρ,u) given in Chen-LeFloch (2003). Finally,for the conditions of P(ρ) given in Chen-LeFloch (2003),applied to the results in Theorems 1 and 2,we obtain the global existence of entropy solutions for the Cauchy problem (1.2) with bounded initial date (1.4).     

Existence of global solutions of delayed differential equations via retract approach

The purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi-global solutions for some classes of delayed functional differential equations. The topological approach known as the topological retract principle is used. Inequalities for coordinates of global solutions are derived as a consequence of used method. Examples illustrate the results.     

Existence of strong solutions and global attractors for the coupled suspension bridge equations

In this paper, we show the existence of the strong solutions for the coupled suspension bridge equations. Furthermore, existence of the strong global attractors is investigated using a new semigroup scheme. Since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.     

Existence of solutions for the equations

We introduce a concept of weak solution for a boundary value problem modelling the interactive motion of a coupled system consisting in a rigid body immersed in a viscous fluid. The fluid, and the solid are contained in a fixed open bounded set of R3. The motion of the fluid is governed by the incompresible Navier-Stokes equations and the standard conservation's laws of linear, and angular momentum rules the dynamics of the rigid body. The time variation of the fluid's domain (due to the motion of the rigid body) is not known apriori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary     

Existence of resurgent solutions for equations with higher-order degeneration

We study the existence of resurgent solutions of differential equations with higher-order degeneration. The proof of the existence of a resurgent solution for the case in which the right-hand side of the equation is a resurgent function is the main result of the present paper.