Let $$H\in C^1\cap W^{2,p}$$ be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field $$b=\nabla ^\perp H$$ . We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T(h) of the closed orbit $$\{H=h\}$$ . Specifically, if $$0<\nu \ll 1$$ is the diffusion...
Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number $$F<2$$ , we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic...
We establish the future nonlinear stability of a large class of FLRW models as solutions to the Einstein-Dust system. We consider the case of a vanishing cosmological constant, which, in particular implies that the expansion rate of the respective models is linear, i.e. has zero acceleration. The resulting spacetimes are future globally regular. These solutions constitute the...
We prove the existence of steady space quasi-periodic stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel $${{\mathbb {R}}}\times [-1,1]$$ . These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal...
We establish $$\textrm{C}^{\infty }$$ -partial regularity results for relaxed minimizers of strongly quasiconvex functionals $$\begin{aligned} \mathscr {F}[u;\Omega ]:=\int _{\Omega }F(\nabla u)\textrm{d}x,\qquad u:\Omega \rightarrow \mathbb {R}^{N}, \end{aligned}$$ subject to a q-growth condition $$|F(z)|\leqq c(1+|z|^{q})$$ , $$z\in \mathbb {R}^{N\times n}$$ , and natural p...
This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As has already been pointed out in the literature, the distance part falls under the classical obstacle...
We show convergence of the Navier–Stokes/Allen–Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our...
This paper addresses the ill-posedness of the classical Rayleigh variational model of conducting charged liquid drops by incorporating the discreteness of the elementary charges. Introducing the model that describes two immiscible fluids with the same dielectric constant, with a drop of one fluid containing a fixed number of elementary charges together with their solvation...
We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side $$f \in L^q$$ for $$q > n+2$$ . In the case of the heat...
In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.
We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and 2d-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples—in particular, we improve in an essentially optimal fashion...
In this paper we consider the steepest descent $$L^2$$ -gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class $$C^2$$ or embedded of class $$W^{2,2}$$ bounding a strictly convex body), the...
We prove that 2-dimensional Q-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and...
A 7-dimensional area-minimizing embedded hypersurface $$M^7$$ will in general have a discrete singular set, and the same is true if M is locally stable provided $${\mathcal {H}}^6(\textrm{sing}M) = 0$$ . We show that if $$M_i^7$$ is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then $$M_i \rightarrow M$$ can limit to a singular $$M^7...
We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.
Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system We consider the critical mass case $$\int _{{\mathbb {R}}^2} u_0(x)\, \textrm{d}x = 8\pi $$ , which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $$u_0^*$$ with mass $$8\pi $$ such that for any initial condition...
We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre...
Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a different approach by maximising directly the dispersive part of the energy functional, while keeping the remaining nonlinear terms fixed with...
We study the homogenization of the Dirichlet problem for the Stokes equations in $$\mathbb {R}^3$$ perforated by m spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order $$m^{-1}$$ , the homogenization limit u is given as the...
We introduce an evolution model à la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, a principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of the existence of a solution to the corresponding parabolic flow, we prove that the stone tends to...
The main result of this work is a homogenization theorem via variational convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumptions of physical growth and non-self-interpenetration. While the obtained energy estimates are rather standard, determining the effective deformation behavior, or in other words, characterizing the weak Sobolev...
The Kelvin–Planck statement of the second law of thermodynamics is a stricture on the nature of heat receipt by any body suffering a cyclic process. It makes no mention of temperature or of entropy. Beginning with a Kelvin–Planck statement of the Second Law, we show that entropy and temperature—in particular, existence of functions that relate the local specific entropy and...
We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $$\Omega $$ is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on $$\Omega $$ . The main feature of these functionals is that the minimality of a domain...