Authors:Verena Bögelein; Frank Duzaar; Paolo Marcellini; Christoph Scheven Pages: 503 - 545 Abstract: In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given by $$\partial_t u^m- {\rm div}\, D_{\xi}\, f(x,Du) =0,$$ with \({m\in (0,\infty )}\) , or more generally with an increasing and piecewise C1 nonlinearity b and a function f depending on u $$\partial_{t}b(u) - {\rm div}\, D_{\xi}\, f(x,u,Du)= -D_u f(x,u,Du).$$ For the function f we merely assume convexity and coercivity, so that, for instance, \({f(x,u,\xi)=\alpha(x) \xi ^p + \beta(x) \xi ^q}\) with 1 < p < q and non-negative coefficients α, β with \({\alpha(x)+\beta(x)\geqq \nu > 0}\) , and \({f(\xi)=\exp(\tfrac12 \xi ^2)}\) are covered. Thus, for functions \({f(x,u,\xi )}\) satisfying only a coercivity assumption from below but very general growth conditions from above, we prove the existence of variational solutions; mean while, if \({f(x,u,\xi )}\) grows naturally when \({\left\vert \xi \right\vert \rightarrow +\infty }\) as a polynomial of order p (for instance in the case of the p-Laplacian operator), then we obtain the existence of solutions in the sense of distributions as well as the existence of weak solutions. Our technique is purely variational and we treat both the cases of bounded and unbounded domains. We introduce a nonlinear version of the minimizing movement approach that could also be useful for the numerics of doubly nonlinear equations. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1221-9 Issue No:Vol. 229, No. 2 (2018)

Authors:Feida Jiang; Neil S. Trudinger Pages: 547 - 567 Abstract: In this paper, we prove the existence of classical solutions to second boundary value problems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics. Our results depend in particular on a priori second derivative estimates recently established by the authors under weak co-dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables, (A3w). We also avoid domain deformations by using the convexity theory of generating functions to construct unique initial solutions for our homotopy family, thereby enabling application of the degree theory for nonlinear oblique boundary value problems. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1222-8 Issue No:Vol. 229, No. 2 (2018)

Authors:Andrea Cianchi; Vladimir G. Maz’ya Pages: 569 - 599 Abstract: Best possible second-order regularity is established for solutions to p-Laplacian type equations with \({p \in (1, \infty)}\) and a square-integrable right-hand side. Our results provide a nonlinear counterpart of the classical L2-coercivity theory for linear problems, which is missing in the existing literature. Both local and global estimates are obtained. The latter apply to solutions to either Dirichlet or Neumann boundary value problems. Minimal regularity on the boundary of the domain is required, although our conclusions are new even for smooth domains. If the domain is convex, no regularity of its boundary is needed at all. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1223-7 Issue No:Vol. 229, No. 2 (2018)

Authors:Bin Cheng; Qiangchang Ju; Steve Schochet Pages: 601 - 625 Abstract: Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1233-5 Issue No:Vol. 229, No. 2 (2018)

Authors:Samuel Punshon-Smith; Scott Smith Pages: 627 - 708 Abstract: This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in \({{L}^{1}}\) . PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1225-5 Issue No:Vol. 229, No. 2 (2018)

Authors:Armin Schikorra Pages: 709 - 788 Abstract: We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, intersect perpendicularly with a support manifold. For example, harmonic maps, or H-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Rivière, that these maps satisfy a system with an antisymmetric potential, from which one can derive the interior regularity of the solution. Avoiding a reflection argument, we show that these maps satisfy along the boundary a system of equations which also exhibits a (nonlocal) antisymmetric potential that combines information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1226-4 Issue No:Vol. 229, No. 2 (2018)

Authors:Nguyen Tien Zung Pages: 789 - 833 Abstract: In this paper we develop a general conceptual approach to the problem of existence of action-angle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions: anything which is preserved by the system is also preserved by the associated torus actions. This approach allows us to obtain, among other things: (a) the shortest and most easy-to-understand conceptual proof of the classical Arnold–Liouville–Mineur theorem; (b) basically all known results in the literature about the existence of action-angle variables in various contexts can be recovered in a unifying way, with simple proofs, using our approach; (c) new results on action-angle variables in many different contexts, including systems on contact manifolds, systems on presymplectic and Dirac manifolds, action-angle variables near singularities, stochastic systems, and so on. Even when there are no natural action variables, our approach still leads to useful normal forms for dynamical systems, which are not necessarily integrable. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1227-3 Issue No:Vol. 229, No. 2 (2018)

Authors:Gang Bao; Yixian Gao; Peijun Li Pages: 835 - 884 Abstract: Consider the scattering of a time-domain acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous air or fluid. This paper concerns the mathematical analysis of such a time-domain acoustic–elastic interaction problem. An exact transparent boundary condition (TBC) is developed to reduce the scattering problem from an open domain into an initial-boundary value problem in a bounded domain. The well-posedness and stability are established for the reduced problem. A priori estimates with explicit time dependence are achieved for the pressure of the acoustic wave field and the displacement of the elastic wave field. Our proof is based on the method of energy, the Lax–Milgram lemma, and the inversion theorem of the Laplace transform. In addition, a time-domain absorbing perfectly matched layer (PML) method is introduced to replace the nonlocal TBC by a Dirichlet boundary condition. A first order symmetric hyperbolic system is derived for the truncated PML problem. The well-posedness and stability are proved. The time-domain PML results are expected to be useful in the computational air/fluid–solid interaction problems. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1228-2 Issue No:Vol. 229, No. 2 (2018)

Authors:Ryan Denlinger Pages: 885 - 952 Abstract: We discuss old and new results on the mathematical justification of Boltzmann’s equation. The classical result along these lines is a theorem which was proven by Lanford in the 1970s. This paper is naturally divided into three parts. I. Classical. We give new proofs of both the uniform bounds required for Lanford’s theorem, as well as the related bounds due to Illner and Pulvirenti for a perturbation of vacuum. The proofs use a duality argument and differential inequalities, instead of a fixed point iteration. II. Strong chaos. We introduce a new notion of propagation of chaos. Our notion of chaos provides for uniform error estimates on a very precise set of points; this set is closely related to the notion of strong (one-sided) chaos and the emergence of irreversibility. III. Supplemental. We announce and provide a proof (in “Appendix A”) of the propagation of partial factorization at some phase-points where complete factorization is impossible. PubDate: 2018-08-01 DOI: 10.1007/s00205-018-1229-1 Issue No:Vol. 229, No. 2 (2018)

Authors:Cleopatra Christoforou; Athanasios E. Tzavaras Pages: 1 - 52 Abstract: We extend the relative entropy identity to the class of hyperbolic–parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful for providing stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an L p framework. It also provides a weak–strong uniqueness theorem for measure valued solutions of the hyperbolic problem. In the second part, the relative entropy identity is developed for the systems of gas dynamics for viscous and heat conducting gases and for the system of thermoviscoelasticity both including viscosity and heat-conduction effects. The dissipation mechanisms and the concentration measures play different roles when applying the method to the general class of hyperbolic–parabolic systems and to the specific examples, and their ramifications are highlighted. PubDate: 2018-07-01 DOI: 10.1007/s00205-017-1212-2 Issue No:Vol. 229, No. 1 (2018)

Authors:Zachary Bradshaw; Tai-Peng Tsai Pages: 53 - 77 Abstract: We construct self-similar solutions to the three dimensional Navier–Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov space \({\dot{B}_{p,\infty}^{3/p-1}}\) where 3 < p < 6. We also construct discretely self-similar solutions for divergence free initial data in \({\dot{B}_{p,\infty}^{3/p-1}}\) for 3 < p < 6 that is discretely self-similar for some scaling factor λ > 1. These results extend those of Bradshaw and Tsai (Ann Henri Poincaré 2016. https://doi.org/10.1007/s00023-016-0519-0) which dealt with initial data in L 3 w since \({L^3_w\subsetneq \dot{B}_{p,\infty}^{3/p-1}}\) for p > 3. We also provide several concrete examples of vector fields in the relevant function spaces. PubDate: 2018-07-01 DOI: 10.1007/s00205-017-1213-1 Issue No:Vol. 229, No. 1 (2018)

Authors:S. Conti; S. Müller; M. Ortiz Pages: 79 - 123 Abstract: We consider a new class of problems in elasticity, referred to as Data-Driven problems, defined on the space of strain-stress field pairs, or phase space. The problem consists of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of approximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within this Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to material data sets that are not graphs. PubDate: 2018-07-01 DOI: 10.1007/s00205-017-1214-0 Issue No:Vol. 229, No. 1 (2018)

Authors:Giacomo Canevari; Antonio Segatti Pages: 125 - 186 Abstract: In this paper we rigorously investigate the emergence of defects on Nematic Shells with a genus different from one. This phenomenon is related to a non-trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete XY system on the shell M, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of Γ-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell M, via the Poincaré–Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects. PubDate: 2018-07-01 DOI: 10.1007/s00205-017-1215-z Issue No:Vol. 229, No. 1 (2018)

Authors:Guowei Yu Pages: 187 - 229 Abstract: In this paper, for the spatial Newtonian 2n-body problem with equal masses, by proving that the minimizers of the action functional under certain symmetric, topological and monotone constraints are collision-free, we found a family of spatial double choreographies, which have the common feature that half of the masses are circling around the z-axis clockwise along a spatial loop, while the motions of the other half of the masses are given by a rotation of the first half around the x-axis by π. Both loops are simple, without any self-intersection, and symmetric with respect to the xz-plane and yz-plane. The set of intersection points between the two loops is non-empty and contained in the xy-plane. The number of such double choreographies grows exponentially as n goes to infinity. PubDate: 2018-07-01 DOI: 10.1007/s00205-018-1216-6 Issue No:Vol. 229, No. 1 (2018)

Authors:Tai-Ping Liu; Shih-Hsien Yu Pages: 231 - 337 Abstract: We study the stability of viscous shock profiles under multi-dimensional perturbation in order to understand the propagation ofmulti-dimensional dispersionwaves over the compressive shock profile. Our analysis is based on the new algorithm of explicitly constructing the Green’s function for the system linearized around the shock profile. We first construct Green’s function for the system linearized around the inviscid profile by matching the two half space problems. The Green’s function around the shock profile is then constructed by iterations. Our approach is of a general nature. We carry out the approach for a simple model possessing the Burgers shocks and wave structure similar to acoustic cones in multi-dimensional gas flows. The approach exhibits a rich phenomenon of wave propagation along and dispersing away from the shock profile. PubDate: 2018-07-01 DOI: 10.1007/s00205-018-1217-5 Issue No:Vol. 229, No. 1 (2018)

Authors:Francisco Gancedo; Eduardo García-Juárez Pages: 339 - 360 Abstract: This paper is about Lions’ open problem on density patches (Lions in Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture series in mathematics and its applications, Clarendon Press, Oxford University Press, New York, 1996): whether or not inhomogeneous incompressible Navier–Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of \({C^{1+\gamma}}\) regularity with \({0 < \gamma < 1}\) in the case of positive density. Furthermore, we go beyond this to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for \({C^{2+\gamma}}\) regularity. PubDate: 2018-07-01 DOI: 10.1007/s00205-018-1218-4 Issue No:Vol. 229, No. 1 (2018)

Authors:Guy Bouchitté; Ilaria Fragalà Pages: 361 - 415 Abstract: We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no duality gap. Further, we provide necessary and sufficient optimality conditions, and we show that our duality principle can be reformulated as a min–max result which is quite useful for numerical implementations. As an example, we illustrate the application of our method to a celebrated free boundary problem. The results were announced in Bouchitté and Fragalà (C R Math Acad Sci Paris 353(4):375–379, 2015). PubDate: 2018-07-01 DOI: 10.1007/s00205-018-1219-3 Issue No:Vol. 229, No. 1 (2018)

Authors:Helmut Abels; Yuning Liu Pages: 417 - 502 Abstract: We consider the sharp interface limit of a coupled Stokes/Allen–Cahn system, when a parameter \({\epsilon > 0}\) that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen–Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen–Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Allen–Cahn operator. Moreover, a careful treatment of the coupling terms is needed. PubDate: 2018-07-01 DOI: 10.1007/s00205-018-1220-x Issue No:Vol. 229, No. 1 (2018)

Authors:J. F. Ganghoffer; P. I. Plotnikov; J. Sokolowski Abstract: The model of volumetric material growth is introduced in the framework of finite elasticity. The new results obtained for the model are presented with complete proofs. The state variables include the deformations, temperature and the growth factor matrix function. The existence of global in time solutions for the quasistatic deformations boundary value problem coupled with the energy balance and the evolution of the growth factor is shown. The mathematical results can be applied to a wide class of growth models in mechanics and biology. PubDate: 2018-06-02 DOI: 10.1007/s00205-018-1259-8

Authors:Wen Deng; Ping Zhang Abstract: Califano and Chiuderi (Phys Rev E 60 (PartB):4701–4707, 1999) conjectured that the energy of an incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimensions provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state (e3, 0). In particular, we prove that for such data, a 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field and e3 decay to zero in both L∞ and L2 norms with explicit rates. We point out that the decay rate in the L2 norm is optimal in sense that this rate coincides with that of the linear system. The main idea of the proof is to exploit Hörmander’s version of the Nash–Moser iteration scheme, which is very much motivated by the seminar papers by Klainerman (Commun Pure Appl Math 33:43–101, 1980, Arch Ration Mech Anal 78:73–98, 1982, Long time behaviour of solutions to nonlinear wave equations. PWN, Warsaw, pp 1209–1215, 1984) on the long time behavior to the evolution equations. PubDate: 2018-06-02 DOI: 10.1007/s00205-018-1265-x