Compactness Arguments Research Articles

Stress Strain Behavior of Compressed Stabilised Earth Blocks using Geogrids

The Existing Situation is employed to concern the soil that is more desirable for compressed stabilised earth block. The behavior of Compressed stabilised earth block (CSEB) on the contrary compressive load, sturdiness, water suction etc. are controlled by the kinds of fine material and stable substance as stickum. The fine material is blended with relevant part of stabilised like fine ashes, cement, coconut fibers and Chemicals are consolidated by labouringly or by using machines. From the review it should be identified that by enumerate Unlike stabilizer in fine aggregate up to a few agreed proportion that rising a practical things of a block. The Specimen is additional strength than typical burnt clay brick. The Soil is checked in a laboratory and considered as a favorable for CSEB on the argument of compactness for the preparation by Montmorillonite Soil. Now it can be decided that the fine aggregate has higher strength than the casual brick but it not please the quality of absorption of water.

On the removal of weak compactness arguments in proof mining

The main observation of this paper is that some sequential weak compactness arguments in Hilbert space theory can be replaced by Heine/Borel compactness arguments (for the strong topology). Even though the latter form of compactness fails in (infinite-dimensional) Hilbert spaces, it nevertheless trivializes under the so-called bounded functional interpretation. As a consequence, the proof mining program of extracting computational bounds from ordinary proofs of mathematics can be applied to modified proofs which use these false Heine/Borel compactness arguments. Additionally, the bounded functional interpretation provides good logical guidance in formulating quantitative versions of analytical statements. We illustrate these claims with three minings. The bounded functional interpretation is here used for the first time in proof mining.

Stationary solutions to coagulation-fragmentation equations

Existence of stationary solutions to the coagulation-fragmentation equation is shown when the coagulation kernel K and the overall fragmentation rate a are given by K(x,y)=xαyβ+xβyα and a(x)=xγ, respectively, with 0≤α≤β≤1, α+β∈[0,1), and γ>0. The proof requires two steps: a dynamical approach is first used to construct stationary solutions under the additional assumption that the coagulation kernel and the overall fragmentation rate are bounded from below by a positive constant. The general case is then handled by a compactness argument.

Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

Nonuniqueness of Weak Solutions to the SQG Equation

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of De Lellis and Székelyhidi in 2012. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. In view of the results of Marchand in 2008, we establish that for the dissipative SQG equation, weak solutions may be constructed in the same function space both via classical weak compactness arguments and via convex integration. © 2019 Wiley Periodicals, Inc.

The marked length spectrum of Anosov manifolds

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stability estimate quantifying how the marked length spectrum control the distance between the metrics. In dimension $2$ we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of $C^\infty$ is finite.

A class of semipositonep-Laplacian problems with a critical growth reaction term

AbstractWe prove the existence of ground state positive solutions for a class of semipositonep-Laplacian problems with a critical growth reaction term. The proofs are established by obtaining crucial uniformC1,αa priori estimates and by concentration compactness arguments. Our results are new even in the semilinear casep= 2.

Homogenization of obstacle problems in Orlicz–Sobolev spaces

We study the homogenization of obstacle problems in Orlicz–Sobolev spaces for a wide class of monotone operators (possibly degenerate or singular) of the p(\cdot) -Laplacian type. Our approach is based on the Lewy–Stampacchia inequalities, which then give access to a compactness argument. We also prove the convergence of the coincidence sets under non-degeneracy conditions.

Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology

In this paper, we apply a rigorous homogenization method based on unfolding operators to a microscopic bidomain model representing the electrical activity of the heart at a cellular level. The heart is represented by an arbitrary open bounded connected domain with smooth boundary and the cardiac cells’ (myocytes) domain is viewed as a periodic region. We start by proving the well posedness of the microscopic problem by using Faedo–Galerkin method and L2-compactness argument on the membrane surface without any restrictive assumptions on the conductivity matrices. Using the unfolding method in homogenization, we show that the sequence of solutions constructed in the microscopic model converges to the solution of the macroscopic bidomain model. Because of the nonlinear ionic function, the proof is based on the surface unfolding method and Kolmogorov compactness argument.

Stable s-minimal cones in ℝ3 are flat for s ~ 1

Abstract We prove that half spaces are the only stable nonlocal s-minimal cones in ℝ 3 {\mathbb{R}^{3}} , for s ∈ ( 0 , 1 ) {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from s = 1 {s=1} . In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

Periodic Solution and Asymptotic Stability for the Magnetohydrodynamic Equations with Inhomogeneous Boundary Condition

We show, using the spectral Galerkin method together with compactness arguments, the existence and uniqueness of the periodic strong solutions for the magnetohydrodynamic-type equations with inhomogeneous boundary conditions. Furthermore, we study the asymptotic stability for the time periodic solution for this system. In particular, when the magnetic field h ( x , t ) is zero, we obtain the existence, uniqueness, and asymptotic behavior of the strong solutions to the Navier–Stokes equations with inhomogeneous boundary conditions.

On the global well‐posedness of the tropical climate model

AbstractIn this paper, we mainly study the Cauchy problem of the tropical climate model in negative‐order Besov spaces. By using the iterative scheme and compactness argument, we first establish the local well‐posedness of the model and then we can extend this local solution to be global if the initial data is sufficient small.

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincare inequality of (Ammann-Grosse-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, well-known theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition.

On the local well-posedness for a full-dispersion Boussinesq system with surface tension

In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplemented with the use of a modified energy in order to close the a priori estimates.

Solutions for critical quasilinear elliptic equations in RN

In this paper, we consider the following critical quasilinear problems in RN: −∑i,j=1NDj(bij(v)Div)+12∑i,j=1Nbij′(v)DivDjv+a(x)v=μ|v|qs−2v+|v|2*s−2v,inRN,v(x)→0 as|x| →∞, where N ≥ 4, bij(v) ∼ 1 + |v|2s−2, s ≥ 1, μ > 0 is a constant, q < 2*, 2*=2NN−2 is the Sobolev critical exponent for the embedding of H1(RN) into Lp(RN), and a(x) is a potential function which is finite and positive and satisfies some decay assumptions. We first use the truncation to overcome the difficulties caused by both the unbounded domain and the critical exponents. Then we perform a change of variables to overcome the difficulty caused by the unboundness of bij. Finally a regularization approach combined with a compact argument helps us to prove the existence of infinitely many solutions.

Finite-time blowup for a Schrödinger equation with nonlinear source term

We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical assumptions $\alpha\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^\alpha U$. In the simplest case, $U(t,x)=(|x|^k-\alpha t)^{-\frac 1\alpha}$ for $t<0$, $x\in {\mathbb R}^N$. For $k$ sufficiently large, $U$ satisfies $|\Delta U|\ll U_t$ close to the blow-up point $(t,x)=(0,0)$, so that it is a suitable approximate solution of the problem. To construct an actual solution $u$ close to $U$, we use energy estimates and a compactness argument.

Well-Posedness of a Cross-Diffusion Population Model with Nonlocal Diffusion

We prove the existence and uniqueness of a solution of a nonlocal cross-diffusion competitive population model for two species. The model may be considered as a version, or even an approximation, of the paradigmatic Shigesada--Kawasaki--Teramoto cross-diffusion model, in which the usual diffusion differential operator is replaced by an integral diffusion operator. The proof of existence of solutions is based on a compactness argument, while the uniqueness of the solution is achieved through a duality technique.

Global zero-relaxation limit of the non-isentropic Euler–Poisson system for ion dynamics

This paper is concerned with smooth solutions of the non-isentropic Euler–Poisson system for ion dynamics. The system arises in the modeling of semi-conductor, in which appear one small parameter, the momentum relaxation time. When the initial data are near constant equilibrium states, with the help of uniform energy estimates and compactness arguments, we rigorously prove the convergence of the system for all time, as the relaxation time goes to zero. The limit system is the drift-diffusion system.

Optimal Distributed Control of a Stochastic Cahn--Hilliard Equation

We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source-term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of an optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analysed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.

Random attractor for the 2D stochastic nematic liquid crystals flows

We consider the long-time behavior for stochastic 2D nematic liquid crystals flows with the velocity field perturbed by an additive noise. The presence of the noises destroys the basic balance law of the nematic liquid crystals flows, so we can not follow the standard argument to obtain uniform a priori estimates for the stochastic flow even in the weak solution space under non-periodic boundary conditions. To overcome the difficulty we use a new technique some kind of logarithmic energy estimates to obtain the uniform estimates which improve the previous result for the orientation field that grows exponentially w.r.t.time t. Considering the existence of random attractor, the common method is to derive uniform a priori estimates in functional space which is more regular than the solution space. We can follow the common method to prove the existence of random attractor in the weak solution space. However, if we consider the existence of random attractor in the strong solution space, it is very difficult and very complicated for such highly non-linear stochastic system with no basic balance law and non-periodic boundary conditions. Here, we use a compactness arguments of the stochastic flow and regularity of the solutions to the stochastic model to obtain the existence of the random attractor in the strong solution space, which implies the support of the invariant measure lies in a more regular space. As far as we know, it is the first article to attack the long-time behavior of stochastic nematic liquid crystals.