An existence theorem for non-homogeneous incompressible fluids

Introduction. In the present paper II we prove existence theorems for weak optimal solutions of nonparametric Lagrange problems with (or without) unilateral constraints. In paper I ([le] of the reference list) we considered arbitrary pairs x(t), u(t) of vector functions, u(t) measurable with values in Em, x(t) absolutely continuous with values in En, and we discussed the existence of the absolute minimum of the functional

In this paper, it is shown that the well known p-harmonic equation in the theory of quasi-conformal mappings also describes some porous medium flows. A weak formulation is made for a free boundary problem with a dam domain. An existence theorem is established for this problem which points the way to a computational approach to the solution.

We prove an existence theorem for weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients. A weak solution of an equation is understood as a weak solution of a stochastic differential inclusion constructed on the basis of the equation. We derive conditions providing the absence of blow-up in weak solutions.

An existence theorem for weak solutions for a class of elliptic partial differential systems in general Orlicz–Sobolev spaces

In this article, we apply the Galerkin approximation method to obtain an existence theorem of weak solutions for 2mth order quasilinear elliptic partial differential resonance equations −Q(u)+f(x,u) =G on a bounded open connected subset Ω of in which the nonlinearity f(x,u) has no growth restriction in u in one of the directions (u→ ∞ and u→ −∞ ) and belongs to o(|u| p−1 ) (p> N/m) in the other, and G∈ [ m,p (Ω )]* may satisfy a Landesman–Lazer condition.

We establish existence theorems for weak solutions of boundary value problems associated with degenerate parabolic operators.

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

In this paper, we study a class of differential inverse variational inequality (for short, DIVI) in finite dimensional Euclidean spaces. Firstly, under some suitable assumptions, we obtain linear growth of the solution set for the inverse variational inequalities. Secondly, we prove existence theorems for weak solutions of the DIVI in the weak sense of Caratheodory by using measurable selection lemma. Thirdly, by employing the results from differential inclusions we establish a convergence result on Euler time dependent procedure for solving the DIVI. Finally, we give a numerical experiment to verify the validity of the algorithm.

Our aim in this paper is to analyze the existence of solutions to a nonlocal elliptic problem involving the fractional Laplacian operator. These operators are utilized to solve an equation defined within a bounded domain in . The operator is a fractional Laplacian , where with the condition and is a non-negative function almost everywhere with respect to the variable , and are real numbers. The article establishes two existence theorems for weak solutions using Tychonoff and Schauder fixed-point theorems. These theorems are formulated based on different hypotheses regarding the parameters and .

Directed fluid sheets and gravity waves in compressible and incompressible fluids

This corrigendum aims to revisit the proof of Lemma 4.1 of [M. Bravin and F. Fanelli, Global existence for non-homogeneous incompressible inviscid fluids in the presence of Ekman pumping, Commun. Contemp. Math. (2025)], which states a continuation critierion for the density-dependent incompressible Euler equations in the class of Besov spaces [Formula: see text], up to the case of critical regularity. The modifications concern only the endpoint case [Formula: see text]. When [Formula: see text], under the same hypotheses of the Lemma, we correct the previous proof, which contained an inexact point. In the critical case [Formula: see text], instead, we need to impose an additional integrability assumption on the gradient of the velocity field. Besides, we improve the argument of the proof to avoid the appearance of a similar assumption on the gradient of the density.

The deformation behavior during axisymmetric upsetting of sintered metals has been studied based on the finite-element method. The investigation on the effects of the initial density distribution, void shape and die friction on the density distribution and punch force during deformation have been conducted. It was found that under low-friction conditions, the initial density distribution affects the deformation geometry and the density distribution. However, the effect of the initial density distribution was found to be negligible under high-friction conditions. The initial density distribution did not affect the punch force or the average density, regardless of the friction conditions. When the force is perpendicular to semi-major axis of elliptical void, it is not only good for densification but also decrease the punch force in forging of porous metal.

The non-uniform medium is regarded as a succession of small-density discontinuities separated by uniform regions. Consideration of the interaction of a shock wave with a weak contact discontinuity gives a first-order relationship between change in shock strength and change in density across the discontinuity, which is integrated to give the shock strength as a function of the initial density of the non-uniform medium in closed form. Due to the passage of the shock, a wave is reflected back through the non-uniform medium, generating in turn a doubly reflected wave which eventually catches up the shock. A complete description of the flow as modified by the first reflected wave is obtained. The modifications to the flow caused by the doubly reflected wave are more difficult to formulate, and a complete description of the flow so modified is not given. The extra difficulty is partly due to the dependence of the doubly reflected wave on the initial density distribution, whereas the motion of the incident shock, and the flow behind it as modified only by the first reflected wave, are found to have the useful property that they are independent of the particular density-distance distribution being considered. Calculations of the total strength of the doubly reflected wave, and the strength of the incident shock when this wave has fully merged with it, have been made for a particular density distribution. A comparison of this calculated strength with the strength of the shock transmitted, after the interaction of a shock wave and a contact discontinuity, suggests that a description of the flow which takes account only of the single and double reflexions is satisfactory, even if the initial density distribution varies considerably.

We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.