Authors:Pekka Koskela; Yi Ru-Ya Zhang Pages: 1 - 14 Abstract: We prove that for a bounded, simply connected domain
\({\Omega \subset {\mathbb{R}^{2}}}\)
, the Sobolev space
\({W^{1,\,\infty}(\Omega)}\)
is dense in
\({W^{1,\,p}(\Omega)}\)
for any
\({1\leqq p < \infty}\)
. Moreover, we show that if
\({\Omega}\)
is Jordan, then
\({C^{\infty}({\mathbb{R}^{2}})}\)
is dense in
\({W^{1,\,p}(\Omega)}\)
for
\({1\leqq p < \infty}\)
. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-0994-y Issue No:Vol. 222, No. 1 (2016)

Authors:Igor Kukavica; Maria Carmela Lombardo; Marco Sammartino Pages: 15 - 45 Abstract: The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be
\({O(\sqrt{\nu})}\)
. The main assumption is spatial analyticity of the initial datum. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-0995-x Issue No:Vol. 222, No. 1 (2016)

Authors:Hongjie Dong; Hong Zhang Pages: 47 - 89 Abstract: In this paper, we derive an interior Schauder estimate for the divergence form elliptic equation
$$D_i (a(x)D_iu) = D_i f_i$$
in
\({\mathbb{R}^2}\)
,where
\({a(x)}\)
and
\({f_i (x)}\)
are piecewise Hölder continuous in a domain containing two touching balls as subdomains. When
\({f_i \equiv 0}\)
and a is piecewise constant, we prove that u is piecewise smoothwith bounded derivatives.This completely answers a question raised by Li andVogelius (Arch Ration Mech Anal 153(2):91–151, 2000) in dimension 2. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-0996-9 Issue No:Vol. 222, No. 1 (2016)

Authors:Christopher P. Hopper Pages: 91 - 141 Abstract: We prove partial regularity for local minimisers of certain strictly quasiconvex integral functionals, over a class of Sobolev mappings into a compact Riemannian manifold, to which such mappings are said to be holonomically constrained. Our approach uses the lifting of Sobolev mappings to the universal covering space, the connectedness of the covering space, an application of Ekeland’s variational principle and a certain tangential
\({\mathbb{A}}\)
-harmonic approximation lemma obtained directly via a Lipschitz approximation argument. This allows regularity to be established directly on the level of the gradient. Several applications to variational problems in condensed matter physics with broken symmetries are also discussed, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-0997-8 Issue No:Vol. 222, No. 1 (2016)

Authors:Zhiwu Lin; Zhengping Wang; Chongchun Zeng Pages: 143 - 212 Abstract: We study the stability of traveling waves of the nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models for this are the Gross–Pitaevskii (GP) equation and the cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type nonlinearity, we construct slow traveling waves and prove their nonlinear instability in any dimension. For dimension two, the non-degeneracy condition is also proved for these slow traveling waves. For general traveling waves without vortices (that is nonvanishing) and with general nonlinearity in any dimension, we find a sharp condition for linear instability. Third, we prove that any 2D traveling wave of (GP) is transversally unstable, and we find the sharp interval of unstable transversal wave numbers. Near unstable traveling waves of all of the above cases, we construct unstable and stable invariant manifolds. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-0998-7 Issue No:Vol. 222, No. 1 (2016)

Authors:Angel Castro; Diego Córdoba; Charles Fefferman; Francisco Gancedo Pages: 213 - 243 Abstract: In this paper we show the existence in finite time of splash singularities for the one-phase Muskat problem. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-0999-6 Issue No:Vol. 222, No. 1 (2016)

Authors:Yong Lu; Zhifei Zhang Pages: 245 - 283 Abstract: This paper focuses on the stability analysis of WKB approximate solutions in geometric optics with the absence of strong transparency conditions under the terminology of Joly, Métivier and Rauch. We introduce a compatible condition and a singular localization method which allows us to prove the stability of WKB solutions over long time intervals. This compatible condition is weaker than the strong transparency condition. The singular localization method allows us to do delicate analysis near resonances. As an application, we show the long time approximation of Klein–Gordon equations by Schrödinger equations in the non-relativistic limit regime. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-1000-4 Issue No:Vol. 222, No. 1 (2016)

Authors:Giovanni P. Galdi Pages: 285 - 315 Abstract: We provide general sufficient conditions for the existence and uniqueness of branching out of a time-periodic family of solutions from steady-state solutions to the two-dimensional Navier-Stokes equations in the exterior of a cylinder. By separating the time-independent averaged component of the velocity field from its oscillatory one, we show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate and distinct function spaces, with the property that the relevant linearized operators become Fredholm of index 0. In this functional setting, the notorious difficulty of 0 being in the essential spectrum entirely disappears and, in fact, it is even meaningless. Our approach is different and, we believe, more natural and simpler than those proposed by previous authors discussing similar questions. Moreover, the latter all fail, when applied to the problem studied here. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-1001-3 Issue No:Vol. 222, No. 1 (2016)

Authors:Irene M. Gamba; Moon-Jin Kang Pages: 317 - 342 Abstract: We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov–Vicsek models that can be considered as non-local, non-linear, Fokker–Planck type equations describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin–Vicsek algorithm as mean-field limit (Bolley et al., Appl Math Lett, 25:339–343, 2012; Degond et al., Math Models Methods Appl Sci 18:1193–1215, 2008), which governs the interactions of stochastic agents moving with a velocity of constant magnitude, that is, the corresponding velocity space for these types of Kolmogorov–Vicsek models is the unit sphere. Our analysis for L
p
estimates and compactness properties take advantage of the orientational interaction property, meaning that the velocity space is a compact manifold. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-1002-2 Issue No:Vol. 222, No. 1 (2016)

Authors:Moon-Jin Kang; Alexis F. Vasseur Pages: 343 - 391 Abstract: We study the contraction properties (up to shift) for admissible Rankine–Hugoniot discontinuities of
\({n\times n}\)
systems of conservation laws endowed with a convex entropy. We first generalize the criterion developed in (Serre and Vasseur, J l’Ecole Polytech 1, 2014), using the spatially inhomogeneous pseudo-distance introduced in (Vasseur, Contemp Math AMS, 2013). Our generalized criterion guarantees the contraction property for extremal shocks of a large class of systems, including the Euler system. Moreover, we introduce necessary conditions for contraction, specifically targeted for intermediate shocks. As an application, we show that intermediate shocks of the two-dimensional isentropic magnetohydrodynamics do not verify any of our contraction properties. We also investigate the contraction properties, for contact discontinuities of the Euler system, for a certain range of contraction weights. None of the results involve any smallness condition on the initial perturbation or on the size of the shock. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-1003-1 Issue No:Vol. 222, No. 1 (2016)

Authors:Scott Armstrong; Antoine Gloria; Tuomo Kuusi Pages: 393 - 426 Abstract: We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-1004-0 Issue No:Vol. 222, No. 1 (2016)

Authors:Stan Alama; Lia Bronsard; Xavier Lamy Pages: 427 - 450 Abstract: We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau–de Gennes model. The nematic is assumed to occupy the exterior of a ball B
r0, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state as
\({ x \to\infty}\)
. We study the minimizers in two different limiting regimes: for balls which are small
\({r_0\ll L^{\frac12}}\)
compared to the characteristic length scale
\({L^{\frac 12}}\)
, and for large balls,
\({r_0\gg L^{\frac12}}\)
. The relationship between the radius and the anchoring strength W is also relevant. For small balls we obtain a limiting quadrupolar configuration, with a “Saturn ring” defect for relatively strong anchoring, corresponding to an exchange of eigenvalues of the Q-tensor. In the limit of very large balls we obtain an axisymmetric minimizer of the Oseen–Frank energy, and a dipole configuration with exactly one point defect is obtained. PubDate: 2016-10-01 DOI: 10.1007/s00205-016-1005-z Issue No:Vol. 222, No. 1 (2016)

Authors:Matthias Hieber; Takahito Kashiwabara Pages: 1077 - 1115 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 DOI: 10.1007/s00205-016-0979-x Issue No:Vol. 221, No. 3 (2016)

Authors:Christophe Lacave; Nader Masmoudi Pages: 1117 - 1160 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 DOI: 10.1007/s00205-016-0980-4 Issue No:Vol. 221, No. 3 (2016)

Authors:Yoshihiro Tonegawa; Neshan Wickramasekera Pages: 1161 - 1222 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 DOI: 10.1007/s00205-016-0981-3 Issue No:Vol. 221, No. 3 (2016)

Authors:Piero D’Ancona; Renato Lucà Pages: 1255 - 1284 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 DOI: 10.1007/s00205-016-0982-2 Issue No:Vol. 221, No. 3 (2016)

Authors:Steinar Evje; Wenjun Wang; Huanyao Wen Pages: 1285 - 1316 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 DOI: 10.1007/s00205-016-0984-0 Issue No:Vol. 221, No. 3 (2016)

Authors:Christian Zillinger Pages: 1449 - 1509 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 DOI: 10.1007/s00205-016-0991-1 Issue No:Vol. 221, No. 3 (2016)

Authors:Mitia Duerinckx; Antoine Gloria Pages: 1511 - 1584 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 DOI: 10.1007/s00205-016-0992-0 Issue No:Vol. 221, No. 3 (2016)

Authors:Songting Luo; Hung V. Tran; Yifeng Yu Pages: 1585 - 1617 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 DOI: 10.1007/s00205-016-0993-z Issue No:Vol. 221, No. 3 (2016)