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)

Abstract: Abstract Various versions of the Navier–Stokes–Allen–Cahn (NSAC), the Navier–Stokes–Cahn–Hilliard (NSCH), and the Navier–Stokes–Korteweg (NSK) equations have been used in the literature to model the dynamics of two-phase fluids. One main purpose of this paper consists in (re-)deriving NSAC, NSCH and NSK from first principles, in the spirit of rational mechanics, for fluids of very general constitutive laws. For NSAC, this deduction confirms and extends a proposal of Blesgen. Regarding NSCH, it continues work of Lowengrub and Truskinovsky and provides the apparently first justified formulation in the non-isothermal case. For NSK, it yields a most natural correction to the formulation by Dunn and Serrin. The paper uniformly recovers as examples various classes of fluids, distinguished according to whether none, one, or both of the phases are compressible, and according to the nature of their co-existence. The latter is captured not only by the mixing energy, but also by a ‘mixing rule’—a constitutive law that characterizes the type of the mixing. A second main purpose of the paper is to communicate the apparently new observation that in the case of two immiscible incompressible phases of different temperature-independent specific volumes, NSAC reduces literally to NSK. This finding may be considered as an independent justification of NSK. An analogous fact is shown for NSCH, which under the same assumption reduces to a new non-local version of NSK. PubDate: 2016-11-30

Abstract: Abstract In this paper, we study the mean field quantum fluctuation dynamics for a system of infinitely many fermions with delta pair interactions in the vicinity of an equilibrium solution (the Fermi sea) at zero temperature, in dimensions d = 2, 3, and prove global well-posedness of the corresponding Cauchy problem. Our work extends some of the recent important results obtained by Lewin and Sabin in [33,34], who addressed this problem for more regular pair interactions. PubDate: 2016-11-29

Abstract: Abstract We establish the inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. In particular, it is employed to prove the existence of weak solutions for the corresponding Cauchy problem. PubDate: 2016-11-25

Abstract: Abstract Two regularity criteria are established to highlight which Littlewood–Paley frequencies play an essential role in possible singularity formation in a Leray–Hopf weak solution to the Navier–Stokes equations in three spatial dimensions. One of these is a frequency localized refinement of known Ladyzhenskaya–Prodi–Serrin-type regularity criteria restricted to a finite window of frequencies, the lower bound of which diverges to \({+\infty}\) as t approaches an initial singular time. PubDate: 2016-11-24

Authors:Raphaël Danchin; Jiang Xu Abstract: Abstract The global existence issue for the isentropic compressible Navier–Stokes equations in the critical regularity framework was addressed in Danchin (Invent Math 141(3):579–614, 2000) more than 15 years ago. However, whether (optimal) time-decay rates could be shown in critical spaces has remained an open question. Here we give a positive answer to that issue not only in the L 2 critical framework of Danchin (Invent Math 141(3):579–614, 2000) but also in the general L p critical framework of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010), Chen et al. (Commun Pure Appl Math 63(9):1173–1224, 2010), Haspot (Arch Ration Mech Anal 202(2):427–460, 2011): we show that under a mild additional decay assumption that is satisfied if, for example, the low frequencies of the initial data are in \({L^{p/2}(\mathbb{R}^{d})}\) , the L p norm (the slightly stronger \({\dot B^0_{p,1}}\) norm in fact) of the critical global solutions decays like \({t^{-d(\frac 1p-\frac14)}}\) for \({t\to+\infty,}\) exactly as firstly observed by Matsumura and Nishida in (Proc Jpn Acad Ser A 55:337–342, 1979) in the case p = 2 and d = 3, for solutions with high Sobolev regularity. Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. PubDate: 2016-11-24 DOI: 10.1007/s00205-016-1067-y

Authors:Seung-Yeal Ha; Tommaso Ruggeri Abstract: Abstract We present a thermodynamically consistent particle (TCP) model motivated by the theory of multi-temperature mixture of fluids in the case of spatially homogeneous processes. The proposed model incorporates the Cucker-Smale (C-S) type flocking model as its isothermal approximation. However, it is more complex than the C-S model, because the mutual interactions are not only “mechanical” but are also affected by the “temperature effect” as individual particles may exhibit distinct internal energies. We develop a framework for asymptotic weak and strong flocking in the context of the proposed model. PubDate: 2016-11-21 DOI: 10.1007/s00205-016-1062-3

Authors:Mahir Hadžić; Steve Shkoller Abstract: Abstract We develop a framework for a unified treatment of well-posedness for the Stefan problem with or without surface tension. In the absence of surface tension, we establish well-posedness in Sobolev spaces for the classical Stefan problem. We introduce a new velocity variable which extends the velocity of the moving free-boundary into the interior domain. The equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature. Solutions to the classical Stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation, in which the moving free-boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem. With our methodology, we simultaneously find the required stability condition for well-posedness and obtain new estimates for the regularity of the moving free-boundary. Finally, we prove that solutions of the Stefan problem with positive surface tension \({\sigma}\) converge to solutions of the classical Stefan problem as \({\sigma \to 0}\) . PubDate: 2016-11-19 DOI: 10.1007/s00205-016-1041-8