In a companion article it was shown in a certain precise sense that, for any thermodynamical theory that respects the Kelvin–Planck second law, the Hahn–Banach theorem immediately ensures the existence of a pair of continuous functions of the local material state—a specific entropy (entropy per mass) and a thermodynamic temperature—that together satisfy the Clausius–Duhem...
In this paper, we introduce Prodi–Serrin like criteria which enable us to provide a priori estimates for the solutions to the spatially homogeneous Landau equation for all classical soft potentials and dimensions $$d \geqq 3$$ . The physical case of Coulomb interaction in dimension $$d=3$$ is included in our analysis; this generalizes the work of Silvestre (J Differ Equ 262:3034...
The Beris–Edwards model of nematic liquid crystals couples an equation for the molecular orientation described by the Q-tensor with a Navier–Stokes type equation with an additional non-Newtonian stress caused by the molecular orientation. Both equations contain a parameter $$\xi \in \mathbb {R}$$ measuring the ratio of tumbling and alignment effects. Previous well-posedness...
We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological conjugation with the archetypal chaotic dynamical system, the Bernoulli shift. After providing infinitely many geometrically distinct and collision...
In this article we propose a novel geometric model to study the motion of a physical flag. In our approach, a flag is viewed as an isometric immersion from the square with values in $$\mathbb {R}^3$$ satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite...
In this article, we study the propagation of defect measures for Schrödinger operators $$-h^2\Delta _g+V$$ on a Riemannian manifold (M, g) of dimension n with V having conormal singularities along a hypersurface Y in the sense that derivatives along vector fields tangential to Y preserve the regularity of V. We show that the standard propagation theorem holds for...
We present a nonlinear stability theory for periodic wave trains in reaction–diffusion systems, which relies on pure $$L^\infty $$ -estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the...
We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. The spectral profiles of these operators describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a...
We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic...
The large-time behavior of solutions to the thin film equation with linear mobility in the complete wetting regime on $$\mathbb {R}^N$$ is examined. We investigate the higher order asymptotics of solutions converging towards self-similar Smyth–Hill solutions under certain symmetry assumptions on the initial data. The analysis is based on a construction of finite-dimensional...
We consider a family of vectorial models for cohesive fracture, which may incorporate $$\textrm{SO}(n)$$ -invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show...
We obtain a new formula for the Loewner energy of Jordan curves on the sphere, which is a Kähler potential for the essentially unique Kähler metric on the Weil–Petersson universal Teichmüller space, as the renormalised energy of moving frames on the two domains of the sphere delimited by the given curve.
We consider the Boltzmann equation in a convex domain with a non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belonging to $$W^{1,p}_x$$ for any $$p<3$$ . We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially quickly as $$t \rightarrow \infty $$ .
The skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in $${\mathbb {R}}^{d+2}$$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge...
In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as “intermittency”, and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of...
We propose a novel weak solution theory for the Mullins–Sekerka equation in dimensions $$d=2$$ and 3, primarily motivated from a gradient flow perspective. Previous existence results on weak solutions due to Luckhaus and Sturzenhecker (Calc. Var. PDE 3, 1995) or Röger (SIAM J. Math. Anal. 37, 2005) left open the inclusion of both a sharp energy dissipation principle and a weak...
We provide a relatively compact proof of the BPHZ theorem for regularity structures of decorated trees in the case where the driving noise satisfies a suitable spectral gap property, as in the Gaussian case. This is inspired by the recent work (Linares et al. in A diagram-free approach to the stochastic estimates in regularity structures, 2021. arXiv:2112.10739 ) in the multi...
We consider the parabolic–elliptic Keller–Segel system in dimensions $$d \geqq 3$$ , which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071–1098, 1999), and is conjectured to be...
We consider the flat flow solution, obtained via a discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from $$C^{1,1}$$ -regular set. We prove the consistency principle, which states that (any) flat flow solution agrees with the classical solution as long as the latter exists. In particular the flat flow solution is unique and smooth up to...
We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem ( $$\textsf{H}\!\!\textsf{K}$$ ), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton–Jacobi equation arising in the dual dynamic formulation of...
In this paper we analyze in detail a few questions related to the theory of functions with bounded p-Hessian–Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the p-Hessian–Schatten total variation, of continuous piecewise linear (CPWL) functions in any space...
We study a variational model for ferronematics in two-dimensional domains, in the “super-dilute” regime. The free energy functional consists of a reduced Landau-de Gennes energy for the nematic order parameter, a Ginzburg–Landau type energy for the spontaneous magnetisation, and a coupling term that favours the co-alignment of the nematic director and the magnetisation. In a...
We construct a family of compact free boundary minimal annuli immersed in the unit ball $$\mathbb {B}^3$$ of $$\mathbb {R}^3$$ , the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical...
We consider the Abelian Yang–Mills–Higgs functional, in the non-self dual scaling, on a complex line bundle over a closed Riemannian manifold of dimension $$n\ge 3$$ . This functional is the natural generalisation of the Ginzburg–Landau model for superconductivity to the non-Euclidean setting. We prove a $$\Gamma $$ -convergence result, in the strongly repulsive limit, on the...