Authors:Yvan Martel; Frank Merle Pages: 1113 - 1160 Abstract: Abstract We construct 2-solitons of the focusing energy-critical nonlinear wave equation in space dimension 5, that is solutions \({u}\) of the equation such that $$u(t) - \left[ W_1(t) + W_2(t)\right] \to 0 \quad \hbox{as } t\to +\infty$$ in the energy space, where \({W_1}\) and \({W_2}\) are Lorentz transforms of the explicit standing soliton \({W(x) = ( 1+ { x ^2}/{15} )^{-3/2}}\) , with any speeds \({\ell_1\neq \ell_2}\) ( \({ \ell_k < 1}\) ). The existence result also holds for the case of \({K}\) -solitons, for any \({K\geq 3}\) , assuming that the speeds \({\ell_k}\) are collinear. The main difficulty of the construction is the strong interaction between the solitons due to the slow algebraic decay of \({W(x)}\) as \({ x \to +\infty}\) . This is in contrast to previous constructions of multi-solitons for other nonlinear dispersive equations (like generalized KdV and nonlinear Schrödinger equations in energy subcritical cases), where the interactions are exponentially small in time due to the exponential decay of the solitons. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1018-7 Issue No:Vol. 222, No. 3 (2016)

Authors:Robert L. Pego; Shu-Ming Sun Pages: 1161 - 1216 Abstract: Abstract We prove an asymptotic stability result for the water wave equations linearized around small solitary waves. The equations we consider govern irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and above by a free surface under the influence of gravity neglecting surface tension. For sufficiently small amplitude waves, with waveform well-approximated by the well-known sech-squared shape of the KdV soliton, solutions of the linearized equations decay at an exponential rate in an energy norm with exponential weight translated with the wave profile. This holds for all solutions with no component in (that is, symplectically orthogonal to) the two-dimensional neutral-mode space arising from infinitesimal translational and wave-speed variation of solitary waves. We also obtain spectral stability in an unweighted energy norm. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1021-z Issue No:Vol. 222, No. 3 (2016)

Authors:V. Ehrlacher; C. Ortner; A. V. Shapeev Pages: 1217 - 1268 Abstract: Abstract Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. This work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed. We formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers. Using this foundation we then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary conditions from nonlinear elasticity. Numerical results confirm the sharpness of the analysis. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1019-6 Issue No:Vol. 222, No. 3 (2016)

Authors:Pearce Washabaugh Pages: 1269 - 1284 Abstract: Abstract We demonstrate that the surface quasi-geostrophic (SQG) equation given by $$\theta_t + \left\langle u, \nabla \theta\right\rangle = 0,\quad \theta = \nabla \times (-\Delta)^{-1/2} u,$$ is the geodesic equation on the group of volume-preserving diffeomorphisms of a Riemannian manifold M in the right-invariant \({\dot{H}^{-1/2}}\) metric. We show by example, that the Riemannian exponential map is smooth and non-Fredholm, and that the sectional curvature at the identity is unbounded of both signs. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1020-0 Issue No:Vol. 222, No. 3 (2016)

Authors:Manuel V. Gnann Pages: 1285 - 1337 Abstract: Abstract We investigate perturbations of traveling-wave solutions to a thin-film equation with quadratic mobility and a zero contact angle at the triple junction, where the three phases liquid, gas and solid meet. This equation can be obtained in lubrication approximation from the Navier–Stokes system of a liquid droplet with a Navier-slip condition at the substrate. Existence and uniqueness have been established by the author together with Giacomelli, Knüpfer and Otto in previous work. As solutions are generically non-smooth, the approach relied on suitably subtracting the leading-order singular expansion at the free boundary. In the present work, we substantially improve this result by showing the regularizing effect of the degenerate-parabolic equation to arbitrary orders of the singular expansion. In comparison to related previous work, our method does not require additional compatibility assumptions on the initial data. The result turns out to be natural in view of the properties of the source-type self-similar profile. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1022-y Issue No:Vol. 222, No. 3 (2016)

Authors:Benedetto Piccoli; Francesco Rossi Pages: 1339 - 1365 Abstract: Abstract The Wasserstein distances W p (p \({\geqq}\) 1), defined in terms of a solution to the Monge–Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou–Brenier formula characterizes the square of the Wasserstein distance W 2 as the infimum of the kinetic energy, or action functional, of all vector fields transporting one measure to the other. Another important property of the Wasserstein distances is the Kantorovich–Rubinstein duality, stating the equality between the distance W 1(μ, ν) of two probability measures μ, ν and the supremum of the integrals in d(μ − ν) of Lipschitz continuous functions with Lipschitz constant bounded by one. An intrinsic limitation of Wasserstein distances is the fact that they are defined only between measures having the same mass. To overcome such a limitation, we recently introduced the generalized Wasserstein distances \({W_p^{a,b}}\) , defined in terms of both the classical Wasserstein distance W p and the total variation (or L 1) distance, see (Piccoli and Rossi in Archive for Rational Mechanics and Analysis 211(1):335–358, 2014). Here p plays the same role as for the classic Wasserstein distance, while a and b are weights for the transport and the total variation term. In this paper we prove two important properties of the generalized Wasserstein distances: (1) a generalized Benamou–Brenier formula providing the equality between \({W_2^{a,b}}\) and the supremum of an action functional, which includes a transport term (kinetic energy) and a source term; (2) a duality à la Kantorovich–Rubinstein establishing the equality between \({W_1^{1,1}}\) and the flat metric. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1026-7 Issue No:Vol. 222, No. 3 (2016)

Authors:Marc Briant; Esther S. Daus Pages: 1367 - 1443 Abstract: Abstract We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in \({L^1_vL^\infty_x(m)}\) , where \({m\sim (1+ v ^k)}\) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an \({L^2-L^\infty}\) theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the \({L^1_vL^\infty_x}\) framework is dealt with for any \({k > k_0}\) , recovering the optimal physical threshold of finite energy \({k_0=2}\) in the particular case of a multi-species hard spheres mixture with the same masses. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1023-x Issue No:Vol. 222, No. 3 (2016)

Authors:Lorena Bociu; Giovanna Guidoboni; Riccardo Sacco; Justin T. Webster Pages: 1445 - 1519 Abstract: Abstract We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid–solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of the existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1024-9 Issue No:Vol. 222, No. 3 (2016)

Authors:Xiaochuan Tian; Qiang Du Pages: 1521 - 1553 Abstract: Abstract We study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators. We present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of Sobolev inequalities such as the nonlocal Poincaré and nonlocal Gagliardo–Nirenberg inequalities. Nonlocal characterizations of high order Sobolev spaces in the spirit of Bourgain–Brezis–Mironescu are provided. Applications of nonlocal calculus of variations to the well-posedness of linear nonlocal models of elastic beams and plates are also considered. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1025-8 Issue No:Vol. 222, No. 3 (2016)

Authors:Walter A. Strauss; Miles H. Wheeler Pages: 1555 - 1580 Abstract: Abstract We consider the angle \({\theta}\) of inclination (with respect to the horizontal) of the profile of a steady two dimensional inviscid symmetric periodic or solitary water wave subject to gravity. Although \({\theta}\) may surpass 30° for some irrotational waves close to the extreme wave, Amick (Arch Ration Mech Anal 99(2):91–114, 1987) proved that for any irrotational wave the angle must be less than 31.15°. Is the situation similar for periodic or solitary waves that are not irrotational? The extreme Gerstner wave has infinite depth, adverse vorticity and vertical cusps (θ = 90°). Moreover, numerical calculations show that even waves of finite depth can overturn if the vorticity is adverse. In this paper, on the other hand, we prove an upper bound of 45° on \({\theta}\) for a large class of waves with favorable vorticity and finite depth. In particular, the vorticity can be any constant with the favorable sign. We also prove a series of general inequalities on the pressure within the fluid, including the fact that any overturning wave must have a pressure sink. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1027-6 Issue No:Vol. 222, No. 3 (2016)

Authors:William C. Troy Pages: 1581 - 1600 Abstract: Abstract We prove the uniqueness of positive ground state solutions of the problem \({ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln( u ) = 0}\) , \({u(r) > 0~\forall r \ge 0}\) , and \({(u(r),u'(r)) \to (0, 0)}\) as \({r \to \infty}\) . This equation is derived from the logarithmic Schrödinger equation \({{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left( u ^{2}\right)}\) , and also from the classical equation \({{\frac {\partial u}{\partial t}} = {\Delta} u +u \left( u ^{p-1}\right) -u}\) . For each \({n \ge 1}\) , a positive ground state solution is \({ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}\) . We combine \({u_{0}(r)}\) with energy estimates and associated Ricatti equation estimates to prove that, for each \({n \in \left[1, 9 \right]}\) , \({u_{0}(r)}\) is the only positive ground state. We also investigate the stability of \({u_{0}(r)}\) . Several open problems are stated. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1028-5 Issue No:Vol. 222, No. 3 (2016)

Authors:Xingfei Xiang Pages: 1601 - 1640 Abstract: Abstract In this paper we study a semilinear system involving the curl operator, which is a limiting form of the Ginzburg–Landau model for superconductors in \({{\mathbb{R}}^3}\) for a large value of the Ginzburg–Landau parameter. We consider the locations of the maximum points of the magnitude of solutions, which are associated with the nucleation of instability of the Meissner state for superconductors when the applied magnetic field is increased in the transition between the Meissner state and the vortex state. For small penetration depth, we prove that the location is not only determined by the tangential component of the applied magnetic field, but also by the normal curvatures of the boundary in some directions. This improves the result obtained by Bates and Pan in Commun. Math. Phys. 276, 571–610 (2007). We also show that the solutions decay exponentially in the normal direction away from the boundary if the penetration depth is small. PubDate: 2016-12-01 DOI: 10.1007/s00205-016-1029-4 Issue No:Vol. 222, No. 3 (2016)

Authors:Patrizio Neff; Bernhard Eidel; Robert J. Martin Pages: 507 - 572 Abstract: Abstract We consider the two logarithmic strain measures $$\begin{array}{ll} {\omega_{\mathrm{iso}}} = {{\mathrm dev}_n {\mathrm log} U} = {{\mathrm dev}_n {\mathrm log} \sqrt{F^TF}} \quad \text{ and } \quad \\ {\omega_{\mathrm{vol}}} = {{\mathrm tr}({\mathrm log} U)} = {{\mathrm tr}({\mathrm log}\sqrt{F^TF})} = {\mathrm log}({\mathrm det} U) \,,\end{array}$$ which are isotropic invariants of the Hencky strain tensor \({\log U}\) , and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group \({{\rm GL}(n)}\) . Here, \({F}\) is the deformation gradient, \({U=\sqrt{F^TF}}\) is the right Biot-stretch tensor, \({\log}\) denotes the principal matrix logarithm, \({\ \cdot \ }\) is the Frobenius matrix norm, \({\rm tr}\) is the trace operator and \({{\text dev}_n X = X- \frac{1}{n} \,{\text tr}(X)\cdot {\mathbb{1}}}\) is the \({n}\) -dimensional deviator of \({X\in{\mathbb {R}}^{n \times n}}\) . This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor \({\varepsilon={\text sym}\nabla u}\) , which is the symmetric part of the displacement gradient \({\nabla u}\) , and reveals a close geometric relation between the classical quadratic isotropic energy potential $$\mu {\ {\text dev}_n {\text sym} \nabla u \ }^2 + \frac{\kappa}{2}{[{\text tr}({\text sym} \nabla u)]}^2 = \mu {\ {\text dev}_n \varepsilon \ }^2 + \frac{\kappa}{2} {[{\text tr} (\varepsilon)]}^2$$ in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy $$\mu {\ {\text dev}_n log U \ }^2 + \frac{\kappa}{2}{[{\text tr}(log U)]}^2 = \mu {\omega_{{\text iso}}^2} + \frac{\kappa}{2}{\omega_{{\text vol}}^2},$$ where \({\mu}\) is the shear modulus and \({\kappa}\) denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor \({R}\) , where \({F=RU}\) is the polar decomposition of \({F}\) . We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1007-x Issue No:Vol. 222, No. 2 (2016)

Authors:Mariana Smit Vega Garcia; Eugen Vărvărucă; Georg S. Weiss Pages: 573 - 601 Abstract: Abstract We consider singularities in the ElectroHydroDynamic equations. In a regime where we are allowed to neglect surface tension, and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non-degeneracy condition, we prove that either the fluid region or the gas region is asymptotically a cusp. Our proofs depend on a combination of monotonicity formulas and a non-vanishing result by Caffarelli and Friedman. As a by-product of our analysis we also obtain a special solution with convex conical air-phase which we believe to be new. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1008-9 Issue No:Vol. 222, No. 2 (2016)

Authors:Ben Stevens Pages: 603 - 730 Abstract: Abstract Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We model the fluids by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density such that the sound speed is positive. We prove that, for a short time, there exists a unique solution of the equations with the same structure. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity which becomes fairly standard after rotating the velocity according to the interface normal. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al. in the setting of a compressible liquid-vacuum interface. Although already considered by Coutand et al. [10] and Lindblad [17], we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1009-8 Issue No:Vol. 222, No. 2 (2016)

Authors:Tong Yang; Hongjun Yu Pages: 731 - 768 Abstract: Abstract We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with \({\gamma\geqq-2}\) . As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate \({t^{-5/4}}\) ) as \({t\to\infty}\) to that of the compressible Navier–Stokes equations for initial data around an equilibrium state. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1010-2 Issue No:Vol. 222, No. 2 (2016)

Authors:Shao-Yuan Huang; Shin-Hwa Wang Pages: 769 - 825 Abstract: Abstract We study global bifurcation curves and the exact multiplicity of positive solutions for the two-point boundary value problem arising in combustion theory: $$\left\{\begin{array}{l}u^{\prime \prime }(x)+\lambda \exp \left(\frac{au}{a+u}\right) =0,\quad -1<x<1, \\ u(-1)=u(1)=0, \end{array}\right.$$ where \({\lambda > 0}\) is the Frank–Kamenetskii parameter and a > 0 is the activation energy parameter. We prove that there exists a critical bifurcation value a 0 \({\approx 4.069}\) such that, on the \({(\lambda, u _{\infty })}\) -plane, the bifurcation curve is S-shaped for \({a > a_{0}}\) and is monotone increasing for \({0 < a \leqq a_{0}}\) . That is, we prove the long-standing conjecture for the one-dimensional perturbed Gelfand problem. We also study, in the \({(a,\lambda, \left \Vert u\right\Vert _{\infty})}\) -space, the shape and structure of the bifurcation surface. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1011-1 Issue No:Vol. 222, No. 2 (2016)

Authors:Oliver Bühler; Jalal Shatah; Samuel Walsh; Chongchun Zeng Pages: 827 - 878 Abstract: Abstract In this work, we consider the mathematical theory of wind generated water waves. This entails determining the stability properties of the family of laminar flow solutions to the two-phase interface Euler equation. We present a rigorous derivation of the linearized evolution equations about an arbitrary steady solution, and, using this, we give a complete proof of the instability criterion of Miles [16]. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air–sea interface). We are thus able to give a unified equation connecting the Kelvin–Helmholtz and quasi-laminar models of wave generation. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1012-0 Issue No:Vol. 222, No. 2 (2016)

Authors:Dominic Breit; Eduard Feireisl; Martina Hofmanová Pages: 895 - 926 Abstract: Abstract We study the asymptotic behavior of the isentropic Navier–Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of a weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic incompressible Navier–Stokes system is identified as the limit problem. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1014-y Issue No:Vol. 222, No. 2 (2016)

Authors:Alexander Kiselev; Xiaoqian Xu Pages: 1077 - 1112 Abstract: Abstract Chemotaxis plays a crucial role in a variety of processes in biology and ecology. In many instances, processes involving chemical attraction take place in fluids. One of the most studied PDE models of chemotaxis is given by the Keller–Segel equation, which describes a population density of bacteria or mold which is attracted chemically to substance they secrete. Solutions of the Keller–Segel equation can exhibit dramatic collapsing behavior, where density concentrates positive mass in a measure zero region. A natural question is whether the presence of fluid flow can affect singularity formation by mixing the bacteria thus making concentration harder to achieve. In this paper, we consider the parabolic-elliptic Keller–Segel equation in two and three dimensions with an additional advection term modeling ambient fluid flow. We prove that for any initial data, there exist incompressible fluid flows such that the solution to the equation stays globally regular. On the other hand, it is well known that when the fluid flow is absent, there exists initial data leading to finite time blow up. Thus the presence of fluid flow can prevent the singularity formation. We discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers. The first class are the relaxation enhancing (RE) flows of (Ann Math:643–674, 2008). These flows are stationary. The second class of flows are the Yao–Zlatos near-optimal mixing flows (Mixing and un-mixing by incompressible flows. arXiv:1407.4163, 2014), which are time dependent. The proof is based on the nonlinear version of the relaxation enhancement construction of (Ann Math:643–674, 2008), and on some variations of the global regularity estimate for the Keller–Segel model. PubDate: 2016-11-01 DOI: 10.1007/s00205-016-1017-8 Issue No:Vol. 222, No. 2 (2016)