We study the inverse problem of determining a Signorini obstacle from boundary measurements for the isotropic elasticity system. We prove that the obstacle can be uniquely determined by a single measurement of displacement and normal stress for the Signorini problem on an open subset of the boundary up to a natural obstruction. In addition to considering the Signorini problem, we...
In this article, we will study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (for example based on finite energy conditions, such as \(L^2\)) does not respect some of the symmetries of the problem, such as Galileo invariance. Our main result, global existence and...
In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri–De Giorgi–Miranda (Arch Rational Mech Anal 32:255–267, 1969) and Simon (Indiana Univ Math J 25:821–855, 1976) to an...
In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by \(|\nabla |^{\alpha }\) for any \(\alpha \in [0, \alpha _0)\) (\(\alpha _0 = \frac{22-8\sqrt{7}}{9} > 0\)). We construct solutions in \(\mathbb {R}^3\times [0,T]\) with a finite \(T>0\) and with...
We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on \(\textrm{BD}(\Omega )\) have weak gradients in \(\textrm{L}_{\textrm{loc}}^{1}(\Omega ;\mathbb {R}^{n\times n})\). This is achieved for the sharp ellipticity range that is presently known to yield \(\textrm{W}_{\textrm{loc}}^{1,1}\)-regularity in the full gradient case...
We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a “droplet state” which is metastable, i.e. persists on a much longer time scale than the time scale of...
We provide a rigorous justification of various kinetic regimes exhibited by the nonlinear Schrödinger equation with an additive stochastic forcing and a viscous dissipation. The importance of such damped-driven models stems from their wide empirical use in studying turbulence for nonlinear wave systems. The force injects energy into the system at large scales, which is then...
We establish the nonlinear stability on a timescale \(O(\varepsilon ^{-2})\) of a linearly, stably stratified rest state in the inviscid Boussinesq system on \(\mathbb {R}^2\). Here, \(\varepsilon >0\) denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation.At the core...
We study the uniqueness of reaction-diffusion steady states in general domains with Dirichlet boundary data. Here we consider “positive” (monostable) reactions. We describe geometric conditions on the domain that ensure uniqueness and we provide complementary examples of nonuniqueness. Along the way, we formulate a number of open problems and conjectures. To derive our results...
We correct the statement of part (2) of Proposition 4.3, correct typographical errors, and clarify a point concerning equation (B.6) for $$\rho _g$$ . Moreover, we provide an improved argument for the proof of the main result: Theorem 6.7. The idea of the proof is to reduce the quasilinear problem to a semilinear problem via the implicit function theorem.
In this paper, we investigate the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady...
We consider the free boundary problem for a 3-dimensional, incompressible, irrotational liquid drop of nearly spherical shape with capillarity. We study the problem from the beginning, extending some classical results from the flat case (capillary water waves) to the spherical geometry: the reduction to a problem on the boundary, its Hamiltonian structure, the analyticity and...
We consider the problem of transfer of energy to high frequencies in a quasilinear Schrödinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a...
We establish the Gauss-Green formula for extended divergence-measure fields (i.e., vector-valued measures whose distributional divergences are Radon measures) over open sets. We prove that, for almost every open set, the normal trace is a measure supported on the boundary of the set. Moreover, for any open set, we provide a representation of the normal trace of the field over the...
In this pages, we consider the p order nonlinear half wave Schrödinger equations $$\begin{aligned} \left( i \partial _{t}+\partial _{x }^2-\left| D_{y}\right| \right) u=\pm |u|^{p-1} u \end{aligned}$$ on the plane \(\mathbb {R}^2\) with \(1<p\le 2\). We prove the global well-posedness of this equation in \(L_x^2 H_y^s(\mathbb {R}^2) \cap H_x^1 L_y^2(\mathbb {R}^2)\)(\(\frac{1}{2...
For a genuinely nonlinear \(2\times 2\) hyperbolic system of conservation laws, assuming that the initial data have a small \(\textbf{L}^\infty \) norm but a possibly unbounded total variation, the existence of global solutions was proven in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like \(t^{-1}\). Motivated by the...
We study a class of radially symmetric Coulomb gas ensembles at inverse temperature \(\beta =2\), for which the droplet consists of a number of concentric annuli, having at least one bounded “gap” G, i.e., a connected component of the complement of the droplet, which disconnects the droplet. Let n be the total number of particles. Among other things, we deduce fine asymptotics as...
We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets...
We consider the \(\phi ^4_1\) measure in an interval of length \(\ell \), defined by a symmetric double-well potential W and inverse temperature \(\beta \). Our results concern its asymptotic behavior in the joint limit \(\beta , \ell \rightarrow \infty \), both in the subcritical regime \(\ell \ll \textrm{e}^{\beta C_W}\) and in the supercritical regime \(\ell \gg \textrm{e...
A static variational model for shape formation in heteroepitaxial crystal growth is considered. The energy functional takes into account surface energy, elastic misfit-energy and nucleation energy of dislocations. A scaling law for the infimal energy is proven. The results quantify the expectation that in certain parameter regimes, island formation or topological defects are...
Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyper-elasticity. However, in its original form the theory of elasticity does not assume the existence of a strain energy function. In this case, called Cauchy elasticity...
We deal with the inverse problem of reconstructing acoustic material properties or/and external sources for the time-domain acoustic wave model. The traditional measurements consist of repeated active (or passive) interrogations, such as the Dirichlet-Neumann map, or point sources with source points varying outside of the domain of interest. It is reported in the existing...
We consider the homogeneous Landau equation in \({\mathbb {R}}^3\) with Coulomb potential and initial data in polynomially weighted \(L^{3/2}\). We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre...
For axisymmetric flows without swirl and compactly supported initial vorticity, we prove the upper bound of \(t^{4/3}\) for the growth of the vorticity maximum, which was conjectured by Childress (Phys. D 237(14-17):1921-1925, 2008) and supported by numerical computations from Childress–Gilbert–Valiant (J. Fluid Mech. 805:1-30, 2016). The key is to estimate the velocity maximum...
We analyze the ground state energy of N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We show that for weak coupling, the ground state energy is asymptotically described by the polaron energy, as proposed by F. Chevy in the physics literature. The polaron energy is the solution of a nonlinear equation involving the Green...