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 Archive for Rational Mechanics and Analysis   [SJR: 3.472]   [H-I: 57]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527    Published by Springer-Verlag  [2335 journals]
• Hamiltonian Formulation for Wave-Current Interactions in Stratified
Rotational Flows
• Abstract: Abstract We show that the Hamiltonian framework permits an elegant formulation of the nonlinear governing equations for the coupling between internal and surface waves in stratified water flows with piecewise constant vorticity.
PubDate: 2016-09-01

• Uniform Regularity and Vanishing Dissipation Limit for the Full
Compressible Navier–Stokes System in Three Dimensional Bounded
Domain
• Abstract: Abstract In the present paper, we study the uniform regularity and vanishing dissipation limit for the full compressible Navier–Stokes system whose viscosity and heat conductivity are allowed to vanish at different orders. The problem is studied in a three dimensional bounded domain with Navier-slip type boundary conditions. It is shown that there exists a unique strong solution to the full compressible Navier–Stokes system with the boundary conditions in a finite time interval which is independent of the viscosity and heat conductivity. The solution is uniformly bounded in $${W^{1,\infty}}$$ and is a conormal Sobolev space. Based on such uniform estimates, we prove the convergence of the solutions of the full compressible Navier–Stokes to the corresponding solutions of the full compressible Euler system in $${L^\infty(0,T; L^2)}$$ , $${L^\infty(0,T; H^{1})}$$ and $${L^\infty([0,T]\times\Omega)}$$ with a rate of convergence.
PubDate: 2016-09-01

• Ill-Posedness of the Hydrostatic Euler and Singular Vlasov Equations
• Abstract: In this paper, we develop an abstract framework to establish ill-posedness, in the sense of Hadamard, for some nonlocal PDEs displaying unbounded unstable spectra. We apply this to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov–Dirac–Benney system.
PubDate: 2016-09-01

• Finite Time Singularity of the Nematic Liquid Crystal Flow in Dimension
Three
• Abstract: Abstract In this paper, we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity. The first example is constructed within the class of axisymmetric solutions, while the second example is constructed for any generic initial data $${(u_0,d_0)}$$ that has sufficiently small energy, and $${d_0}$$ has a nontrivial topology.
PubDate: 2016-09-01

• Impermeability Through a Perforated Domain for the Incompressible two
dimensional Euler Equations
• Abstract: Abstract We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $${\varepsilon}$$ separated by distances $${d_{\varepsilon}}$$ and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of $${\frac{d_{\varepsilon}}\varepsilon}$$ when $${\varepsilon}$$ goes to zero. If $${\frac{d_{\varepsilon}}\varepsilon \to \infty}$$ , then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, $${\frac{d_{\varepsilon}}\varepsilon \to 0}$$ , then the fluid cannot penetrate the porous region, namely, the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of $${\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}}$$ where $${\gamma \in (0,\infty]}$$ is related to the geometry of the lateral boundaries of the obstacles. If $${\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty}$$ , then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions; in particular, it is equal to $${\varepsilon^{3}}$$ for balls.
PubDate: 2016-09-01

• Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with
Convex Growth
• Abstract: Abstract We consider the well-trodden ground of the problem of the homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has p-growth from below (with p > d, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals.
PubDate: 2016-09-01

• On the Regularity Set and Angular Integrability for the
Navier–Stokes Equation
• Abstract: Abstract We investigate the size of the regular set for suitable weak solutions of the Navier–Stokes equation, in the sense of Caffarelli–Kohn–Nirenberg (Commun Pure Appl Math 35:771–831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space $${\{t > 0\}}$$ in an appropriate limit. In particular, we obtain that if the $${L^{2}}$$ norm with weight $${ x ^{-\frac12}}$$ of the data tends to 0, the regular set invades $${\{t > 0\}}$$ ; this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).
PubDate: 2016-09-01

• Linear Inviscid Damping for Monotone Shear Flows in a Finite Periodic
Channel, Boundary Effects, Blow-up and Critical Sobolev Regularity
• Abstract: Abstract In a previous article (Zillinger, Linear inviscid damping for monotone shear flows, 2014), we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution as $${t \rightarrow \infty}$$ . As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a finite periodic channel, where high regularity would be essential to control nonlinear effects.
PubDate: 2016-09-01

• Global Well-Posedness and Decay Rates of Strong Solutions to a
Non-Conservative Compressible Two-Fluid Model
• Abstract: Abstract In this paper, we consider a compressible two-fluid model with constant viscosity coefficients and unequal pressure functions $${P^+\neq P^-}$$ . As mentioned in the seminal work by Bresch, Desjardins, et al. (Arch Rational Mech Anal 196:599–629, 2010) for the compressible two-fluid model, where $${P^+=P^-}$$ (common pressure) is used and capillarity effects are accounted for in terms of a third-order derivative of density, the case of constant viscosity coefficients cannot be handled in their settings. Besides, their analysis relies on a special choice for the density-dependent viscosity [refer also to another reference (Commun Math Phys 309:737–755, 2012) by Bresch, Huang and Li for a study of the same model in one dimension but without capillarity effects]. In this work, we obtain the global solution and its optimal decay rate (in time) with constant viscosity coefficients and some smallness assumptions. In particular, capillary pressure is taken into account in the sense that $${\Delta P=P^+ - P^-=f\neq 0}$$ where the difference function $${f}$$ is assumed to be a strictly decreasing function near the equilibrium relative to the fluid corresponding to $${P^-}$$ . This assumption plays an key role in the analysis and appears to have an essential stabilization effect on the model in question.
PubDate: 2016-09-01

• Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi
Equations
• Abstract: Abstract We look at the effective Hamiltonian $${\overline{H}}$$ associated with the Hamiltonian $${H(p,x)=H(p)+V(x)}$$ in the periodic homogenization theory. Our central goal is to understand the relation between $${V}$$ and $${\overline{H}}$$ . We formulate some inverse problems concerning this relation. Such types of inverse problems are, in general, very challenging. In this paper, we discuss several special cases in both convex and nonconvex settings.
PubDate: 2016-09-01

• The Blow Up Method for Brakke Flows: Networks Near Triple Junctions
• Abstract: Abstract We introduce a parabolic blow-up method to study the asymptotic behavior of a Brakke flow of planar networks (that is a 1-dimensional Brakke flow in a two dimensional region) weakly close in a space-time region to a static multiplicity 1 triple junction J. We show that such a network flow is regular in a smaller space-time region, in the sense that it consists of three curves coming smoothly together at a single point at 120 $${^{\circ}}$$ angles, staying smoothly close to J and moving smoothly. Using this result and White’s stratification theorem, we deduce that whenever a Brakke flow of networks in a space-time region $${{\mathcal {R}}}$$ has no static tangent flow with density $${{\geqq}2}$$ , there exists a closed subset $${{\Sigma \subset {\mathcal {R}}}}$$ of parabolic Hausdorff dimension at most 1 such that the flow is classical in $${{\mathcal {R}}\backslash\Sigma}$$ , that is near every point in $${{\mathcal {R}}\backslash\Sigma}$$ , the flow, if non-empty, consists of either an embedded curve moving smoothly or three embedded curves meeting smoothly at a single point at 120 $${^{\circ}}$$ angles and moving smoothly. In particular, such a flow is classical at all times except for a closed set of times of ordinary Hausdorff dimension at most $${\frac{1}{2}}$$ .
PubDate: 2016-09-01

• Global Strong Well-Posedness of the Three Dimensional Primitive Equations
in $${L^p}$$ L p -Spaces
• Abstract: Abstract In this article, an $${L^p}$$ -approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data $${a \in [X_p,D(A_p)]_{1/p}}$$ provided $${p \in [6/5,\infty)}$$ . To this end, the hydrostatic Stokes operator $${A_p}$$ defined on $${X_p}$$ , the subspace of $${L^p}$$ associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing $${p}$$ large, one obtains global well-posedness of the primitive equations for strong solutions for initial data $${a}$$ having less differentiability properties than $${H^1}$$ , hereby generalizing in particular a result by Cao and Titi (Ann Math 166:245–267, 2007) to the case of non-smooth initial data.
PubDate: 2016-09-01

• A New Contraction Family for Porous Medium and Fast Diffusion Equations
• Abstract: Abstract In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation.
PubDate: 2016-08-01

• On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular
Domains
• Abstract: Abstract In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The “free vibrating membrane” corresponds to the Neumann–Laplace operator in bounded plane domains. In this paper we obtain estimates for the first non-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This class includes convex domains, John domains etc. On the basis of our estimates we conjecture that the eigenvalues of the Neumann– Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates which is based on weighted Poincaré–Sobolev inequalities, obtained by the authors recently.
PubDate: 2016-08-01

• A Parabolic Problem with a Fractional Time Derivative
• Abstract: Abstract We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi–Nash–Moser Hölder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.
PubDate: 2016-08-01

• Exponential Stability of Slowly Decaying Solutions to the
Kinetic-Fokker-Planck Equation
• Abstract: The aim of the present paper is twofold: We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the “mild perturbation” part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series. We apply this method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap estimates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W 1.
PubDate: 2016-08-01

• On the Impossibility of Finite-Time Splash Singularities for Vortex Sheets
• Abstract: Abstract In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet evolution, that is for the two-phase incompressible Euler equations. We prove this by contradiction; we assume that a splash singularity does indeed occur in finite time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allow us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, showing that our assumption of a finite-time splash singularity was false.
PubDate: 2016-08-01

• Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body
• Abstract: Abstract The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal velocity in the upstream, there exists a critical value $${\rho_{\rm cr}}$$ such that if the incoming density in the upstream is larger than $${\rho_{\rm cr}}$$ , then there exists a subsonic flow past a wall. Furthermore, $${\rho_{\rm cr}}$$ is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than $${\rho_{\rm cr}}$$ . The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states.
PubDate: 2016-08-01

• Local Existence of Weak Solutions to Kinetic Models of Granular Media
• Abstract: Abstract We prove in any dimension $${d \geqq 1}$$ a local in time existence of weak solutions to the Cauchy problem for the kinetic equation of granular media, $$\partial_t f+v\cdot \nabla_x f = {div}_v[f(\nabla W *_v f)]$$ when the initial data are nonnegative, integrable and bounded functions with compact support in velocity, and the interaction potential $${W}$$ is a $${C^2({\mathbb{R}}^d)}$$ radially symmetric convex function. Our proof is constructive and relies on a splitting argument in position and velocity, where the spatially homogeneous equation is interpreted as the gradient flow of a convex interaction energy with respect to the quadratic Wasserstein distance. Our result generalizes the local existence result obtained by Benedetto et al. (RAIRO Modél Math Anal Numér 31(5):615–641, 1997) on the one-dimensional model of this equation for a cubic power-law interaction potential.
PubDate: 2016-08-01

• Magnetic WKB Constructions
• Abstract: Abstract This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in the presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman–Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born–Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions.We also investigate explicit examples for which we can improve our general theorem: global WKB expansions, quasi-optimal Agmon estimates and upper bound of the tunelling effect (in symmetric cases).We also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades.
PubDate: 2016-08-01

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