Abstract: We are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: $$\partial_2u -P( a(u)\partial_1^2u + \sigma(u)f ) =0,$$ where P is the projection on mean-zero functions, and f is a distribution which is only controlled in the low regularity norm of \({ C^{\alpha-2}}\) for \({\alpha > \frac{2}{3}}\) on the parabolic Hölder scale. The example we have in mind is a random forcing f and our assumptions allow, for example, for an f which is white in the time variable x2 and only mildly coloured in the space variable x1; any spatial covariance operator \({(1 + \partial_1 )^{-\lambda_1 }}\) with \({\lambda_1 > \frac13}\) is admissible. On the deterministic side we obtain a \({C^\alpha}\) -estimate for u, assuming that we control products of the form \({v\partial_1^2v}\) and vf with v solving the constant-coefficient equation \({\partial_2 v-a_0\partial_1^2v=f}\) . As a consequence, we obtain existence, uniqueness and stability with respect to \({(f, vf, v \partial_1^2v)}\) of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing f using stochastic arguments. For this we extend the treatment of the singular product \({\sigma(u)f}\) via a space-time version of Gubinelli’s notion of controlled rough paths to the product \({a(u)\partial_1^2u}\) , which has the same degree of singularity but is more nonlinear since the solution u appears in both factors. In fact, we develop a theory for the linear equation \({\partial_t u - P(a\partial_1^2 u +\sigma f)=0}\) with rough but given coefficient fields a and \({\sigma}\) and then apply a fixed point argument. The PDE ingredient mimics the (kernel-free) Safonov approach to ordinary Schauder theory. PubDate: 2019-05-01

Abstract: We study the beach problem for water waves. The case we consider is a compact fluid domain, where the free surface intersects the bottom along an edge, with a non-zero contact angle. Using elliptic estimates in domains with edges and a new equation on the Taylor coefficient, we establish a priori estimates for angles smaller than a dimensional constant. Local existence will be derived in a following paper. PubDate: 2019-05-01

Abstract: In this paper, we are concerned with the well-posedness theory of steady incompressible jet flow in a de Laval type nozzle with given end pressure at the outlet. The main results show that for any given incoming mass flux Q > 0 in the upstream and end pressure at the outlet, there exists an admissible interval to the Bernoulli’s constant, if the Bernoulli’s constant lies in the interval, there exists a unique smooth incompressible jet flow issuing from the nozzle. Moreover, the free boundary of the jet flow initiates smoothly from the surface of the divergent area of the de Laval nozzle. In particular, it is shown that the initial point of the free boundary lies behind the throat of the de Laval nozzle wall and varies continuously and monotonically with respect to the Bernoulli’s constant. As a direct corollary, imposing initial point of the free boundary on the divergent part of nozzle wall, there exists a unique incompressible jet for given incoming mass flux Q > 0 and pressure Pe at the outlet. This work is inspired by the significant works (Alt et al. in Commun Pure Appl Math 35:29–68, 1982; Arch Rational Mech Anal 81:97–149, 1983) for the incompressible jet flow imposing the initial point of the free boundary at the endpoint of the nozzle. PubDate: 2019-05-01

Abstract: The well-posedness of classical solutions with finite energy to the compressible Navier–Stokes equations (CNS) subject to arbitrarily large and smooth initial data is a challenging problem. In the case when the fluid density is away from vacuum (strictly positive), this problem was first solved for the CNS in either one-dimension for general smooth initial data or multi-dimension for smooth initial data near some equilibrium state (that is, small perturbation) (Antontsev et al. in Boundary value problems in mechanics of nonhomogeneous fluids, North-Holland Publishing Co., Amsterdam, 1990; Kazhikhov in Sibirsk Mat Zh 23:60–64, 1982; Kazhikhov et al. in Prikl Mat Meh 41:282–291, 1977; Matsumura and Nishida in Proc Jpn Acad Ser A Math Sci 55:337–342, 1979, J Math Kyoto Univ 20:67–104, 1980, Commun Math Phys 89:445–464, 1983). In the case that the flow density may contain a vacuum (the density can be zero at some space-time point), it seems to be a rather subtle problem to deal with the well-posedness problem for CNS. The local well-posedness of classical solutions containing a vacuum was shown in homogeneous Sobolev space (without the information of velocity in L2-norm) for general regular initial data with some compatibility conditions being satisfied initially (Cho et al. in J Math Pures Appl (9) 83:243–275, 2004; Cho and Kim in J Differ Equ 228:377–411, 2006, Manuscr Math 120:91–129, 2006; Choe and Kim in J Differ Equ 190:504–523 2003), and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations (Huang et al. in Commun Pure Appl Math 65:549–585, 2012). However, it was shown that any classical solutions to the compressible Navier–Stokes equations in finite energy (inhomogeneous Sobolev) space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported (Xin in Commun Pure Appl Math 51:229–240, 1998). In this paper, we investigate the well-posedess of classical solutions to the Cauchy problem of Navier–Stokes equations, and prove that the classical solution with finite energy does not exist in the inhomogeneous Sobolev space for any short time under some natural assumptions on initial data near the vacuum. This implies, in particular, that the homogeneous Sobolev space is as crucial as studying the well-posedness for the Cauchy problem of compressible Navier–Stokes equations in the presence of a vacuum at far fields even locally in time. PubDate: 2019-05-01

Abstract: We are concerned with the nonlinear stability of vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic vortex sheets is obtained by analyzing the roots of the Lopatinskiĭ determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing the error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic vortex sheets under small initial perturbations by a Nash–Moser iteration scheme. PubDate: 2019-05-01

Abstract: The Merriman–Bence–Osher (MBO) scheme, also known as diffusion generated motion or thresholding, is an efficient numerical algorithm for computing mean curvature flow (MCF). It is fairly well understood in the case of hypersurfaces. This paper establishes the first convergence proof of the scheme in codimension two. We concentrate on the case of the curvature motion of a filament (curve) in \({{\mathbb{R}}^3}\) . Our proof is based on a new generalization of the minimizing movements interpretation for hypersurfaces (Esedoglu–Otto ’15) by means of an energy that approximates the Dirichlet energy of the state function. As long as a smooth MCF exists, we establish uniform energy estimates for the approximations away from the smooth solution and prove convergence towards this MCF. The current result that holds in codimension two relies in a very crucial manner on a new sharp monotonicity formula for the thresholding energy. This is an improvement of an earlier approximate version. PubDate: 2019-05-01

Abstract: We revisit the question of global regularity for the Patlak–Keller–Segel (PKS) chemotaxis model. The classical 2D parabolic-elliptic model blows up for initial mass \({M > 8\pi}\) . We consider a more realistic scenario which takes into account the flow of the ambient environment induced by harmonic potentials, and thus retain the identical elliptic structure as in the original PKS. Surprisingly, we find that already the simplest case of linear stationary vector field, \({Ax^\top}\) , with large enough amplitude \({A}\) , prevents chemotactic blow-up. Specifically, the presence of such an ambient fluid transport creates what we call a ‘fast splitting scenario’, which competes with the focusing effect of aggregation so that ‘enough mass’ is pushed away from concentration along the \({x_1}\) -axis, thus avoiding a finite time blow-up, at least for \({M < 16\pi}\) . Thus, the enhanced ambient flow doubles the amount of allowable mass which evolve to global smooth solutions. PubDate: 2019-05-01

Abstract: The evolutionary Eikonal equation is a Hamilton–Jacobi equation with Hamiltonian H(P) = P , which is not strictly convex nor smooth. The regularizing effect of Hamiltonian for the Eikonal equation is much weaker than that of strictly convex Hamiltonians, therefore leading to new phenomena. In this paper, we study the set of singularity points of solutions in the upper half space for C1 or C2 initial data, with emphasis on the countability of connected components of the set. The regularity of solutions in the complement of the set of singularity points is also obtained. PubDate: 2019-05-01

Abstract: We consider the electrostatic Born–Infeld energy $$\int_{\mathbb{R}^N}\left(1-{\sqrt{1- \nabla u ^2}}\right)\, {\rm d}x -\int_{\mathbb{R}^N}\rho u\, {\rm d}x,$$ where \({\rho \in L^{m}(\mathbb{R}^N)}\) is an assigned charge density, \({m \in [1,2_*]}\) , \({2_*:=\frac{2N}{N+2}}\) , \({N\geq 3}\) . We prove that if \({\rho \in L^q(\mathbb{R}^N) }\) for \({q > 2N}\) , the unique minimizer \({u_\rho}\) is of class \({W_{loc}^{2,2}(\mathbb{R}^N)}\) . Moreover, if the norm of \({\rho}\) is sufficiently small, the minimizer is a weak solution of the associated PDE $$\label{eq:BI-abs}-\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1- \nabla u ^2}}\right)= \rho \quad\hbox{in }\mathbb{R}^N,\quad \quad \quad \mathcal{(BI)}$$ with the boundary condition \({\lim_{ x \to\infty}u(x)=0}\) , and it is of class \({C^{1,\alpha}_{loc}(\mathbb{R}^N)}\) for some \({\alpha \in (0,1)}\) . PubDate: 2019-05-01

Abstract: In the present paper we study stochastic homogenization for reaction–diffusion equations with stationary ergodic reactions (including periodic). We first show that under suitable hypotheses, initially localized solutions to the PDE asymptotically become approximate characteristic functions of a ballistically expanding Wulff shape. The next crucial component is the proper definition of relevant front speeds and the subsequent establishment of their existence. We achieve the latter by finding a new relation between the front speeds and the Wulff shape, provided the Wulff shape does not have corners. Once front speeds are proved to exist in all directions, by the above means or otherwise, we are able to obtain general stochastic homogenization results, showing that large space–time evolution of solutions to the PDE is governed by a simple deterministic Hamilton–Jacobi equation whose Hamiltonian is given by these front speeds. Our results are new even for periodic reactions, particularly of ignition type. We primarily consider the case of non-negative reactions but we also extend our results to the more general PDE \({u_{t}= F(D^2 u,\nabla u,u,x,\omega)}\) , as long as its solutions satisfy some basic hypotheses including positive lower and upper bounds on spreading speeds in all directions and a sub-ballistic bound on the width of the transition zone between the two equilibria of the PDE. PubDate: 2019-05-01

Abstract: We prove the existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions–Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier–Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via an unconventional approximation using the Lagrangian formulation, truncations, and a stability result of trajectories for rough velocity fields. PubDate: 2019-05-01

Abstract: We investigate the scaling of the ground state energy and optimal domain patterns in thin ferromagnetic films with strong uniaxial anisotropy and the easy axis perpendicular to the film plane. Starting from the full three-dimensional micromagnetic model, we identify the critical scaling for which the transition from single domain to multidomain ground states such as bubble or maze patterns occurs as the film thickness goes to zero and the lateral extent goes to infinity. Furthermore, we analyze the asymptotic behavior of the energy in these two asymptotic regimes. In the single domain regime, the energy Γ-converges towards a much simpler two-dimensional and local model. In the multidomain regime, we derive the scaling of the minimal energy and deduce a scaling law for the typical domain size. PubDate: 2019-05-01

Abstract: In this article we construct global solutions to a simplified Ericksen–Leslie system on \({\mathbb{R}^3}\) . The constructed solutions are twisted and periodic along the x3-axis with period \({d = 2\pi \big/ \mu}\) . Here \({\mu > 0}\) is the twist rate and d is the distance between two planes which are parallel to the x1x2-plane. Liquid crystal material is placed in the region enclosed by these two planes. Given a well-prepared initial data, our solutions exist classically for all \({t \in (0, \infty)}\) . However, these solutions become singular at all points on the x3-axis and escape into third dimension exponentially while \({t \rightarrow \infty}\) . An optimal blow up rate is also obtained. PubDate: 2019-04-01

Abstract: We study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous Hamiltonian. We find an explicit formula for the Hamiltonian, prove that the solutions are unique, and prove that the limiting free boundary has a facets in every rational direction. Our choice of problem presents difficulties that require the development of a new uniqueness proof for certain free boundary problems. The problem is motivated by physical experiments involving liquid drops on patterned solid surfaces. PubDate: 2019-04-01

Abstract: We introduce a notion of global weak solution to the Navier–Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces \({\dot{B}^{-1+\frac{3}{p}}_{p,\infty}}\) , p > 3. These solutions satisfy a certain stability property with respect to the weak- \({\ast}\) convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest. PubDate: 2019-04-01

Abstract: We explore the local existence and properties of classical weak solutions to the initial-boundary value problem for a class of quasilinear equations of elastodynamics in one space dimension with a non-convex stored-energy function, a model of phase transitions in elastic bars proposed by Ericksen (J Elast 5(3–4):191–201,1975). The instantaneous phase separation and formation of microstructures of such solutions are observed for all smooth initial data with initial strain having its range that overlaps with the phase transition zone of the Piola–Kirchhoff stress. Moreover, we can select those solutions in a way that their phase gauges are close to a certain number inherited from a modified hyperbolic problem and thus give rise to an internal strain–stress hysteresis loop. As a byproduct, we prove the existence of a measure-valued solution to the problem that is generated by a sequence of weak solutions but not a weak solution itself. It is also shown that the problem admits a local weak solution for all smooth initial data and local weak solutions that are smooth for a short period of time and exhibit microstructures thereafter for certain smooth initial data. PubDate: 2019-04-01

Abstract: We consider the gradient flow evolution of a phase-field model for crystal dislocations in a single slip system in the presence of forest dislocations. The model is based on a Peierls–Nabarro type energy penalizing non-integer slip and elastic stress. Forest dislocations are introduced as a perforation of the domain by small disks where slip is prohibited. The \({\Gamma}\) -limit of this energy was deduced by Garroni and Müller (SIAM J Math Anal 36(6):1943–1964, 2005, Arch Ration Mech Anal 181(3):535–578, 2006). Our main result shows that the gradient flows of these \({\Gamma}\) -convergent energy functionals do not approach the gradient flow of the limiting energy. Indeed, the gradient flow dynamics remains a physically reasonable model in the case of non-monotone loading. Our proofs rely on the construction of explicit sub- and super-solutions to a fractional Allen–Cahn equation on a flat torus or in the plane, with Dirichlet data on a union of small discs. The presence of these obstacles leads to an additional friction in the viscous evolution which appears as a stored energy in the \({\Gamma}\) -limit, but it does not act as a driving force. Extensions to related models with soft pinning and non-viscous evolutions are also discussed. In terms of physics, our results explain how in this phase field model the presence of forest dislocations still allows for plastic as opposed to only elastic deformation. PubDate: 2019-04-01

Abstract: In this paper, we consider a confined physical scenario to prove the global existence of smooth solutions with bounded density and finite energy for the inviscid incompressible porous media (IPM) equation. The result is proved using the stability of stratified solutions, combined with an additional structure of our initial perturbation, which allows us to get rid of the boundary terms in the energy estimates. PubDate: 2019-04-01

Abstract: In this note we combine the “spin-argument” from Kitavtsev et al. (Proc R Soc Edinb Sect A Mater 147(5):1041–1089, 2017) and the n-dimensional incompatible, one-well rigidity result from Lauteri and Luckhaus (An energy estimate for dislocation configurations and the emergence of Cosserat-type structures in metal plasticity, 2016), in order to infer a new proof for the compactness of discrete multi-well energies associated with the modelling of surface energies in certain phase transitions. Mathematically, a main novelty here is the reduction of the problem to an incompatible one-well problem. The presented argument is very robust and applies to a number of different physically interesting models, including for instance phase transformations in shape-memory materials but also anti-ferromagnetic transformations or related transitions with an “internal” microstructure on smaller scales. PubDate: 2019-04-01

Abstract: Consider electromagnetic waves in two-dimensional honeycomb structured media, whose constitutive laws have the symmetries of a hexagonal tiling of the plane. The properties of transverse electric polarized waves are determined by the spectral properties of the elliptic operator \({\mathcal{L}^{A}=-\nabla_{\bf x}\cdot A({\bf x}) \nabla_{\bf x}}\) , where A(x) is \({{\Lambda}_h}\) -periodic ( \({{\Lambda}_h}\) denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is \({\mathcal{P}\mathcal{C}}\) -invariant ( \({A({\bf x})=\overline{A(-{\bf x})}}\) ) and \({120^\circ}\) rotationally invariant ( \({A(R^*{\bf x})=R^*A({\bf x})R}\) , where R is a \({120^\circ}\) rotation in the plane). A summary of our results is as follows: (a) For generic honeycomb structured media, the band structure of \({\mathcal{L}^{A}}\) has Dirac points, i.e. conical intersections between two adjacent Floquet–Bloch dispersion surfaces; (b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose wave-envelope, on long time scales, is governed by an effective two-dimensional time-dependent system of massless Dirac equations; (c) Dirac points are unstable to arbitrary small perturbations which break either \({\mathcal{C}}\) (complex-conjugation) symmetry or \({\mathcal{P}}\) (inversion) symmetry; (d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell’s equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term; (e) These results imply the existence of unidirectional propagating edge states for two classes of time-reversal invariant media in which \({\mathcal{C}}\) symmetry is broken: magneto-optic media and bi-anisotropic media. PubDate: 2019-04-01