Blockchain Papers

Mitsuo Higaki, Jiajiang Liao, Franck Sueur

The question at stake in Lagrangian controllability is whether one can move a patch of fluid particles to a target location by means of remote action in a given time interval. In the last two decades, positive results have been obtained both for the incompressible Euler and Navier-Stokes equations. However, for the latter, the case where the fluid is contained within domains bounded by solid boundaries with the no-slip condition has not been addressed, with respect to the difficulty caused by viscous boundary layers. In this paper, we investigate the Lagrangian controllability of viscous incompressible fluid in perforated domains for which the fraction of volume occupied by the holes is sufficiently small. Moreover, we quantitatively distinguish situations depending on the parameters for holes (diameter and distance) and for fluid (size of the initial data). Our approach relies on recent results on homogenization for evolutionary problems and on weak-strong stability estimates in measure of flows, alongside classical results on Runge-type approximations for elliptic equations and on Cauchy-Kowalevsky-type theorems for equations with analytic coefficients. Here, homogenization refers to the vanishing viscosity limit outside a porous medium, where (after scaling in time) the Navier-Stokes equations are homogenized to the Euler or Darcy equations. Indeed, in the proof, we act on the Navier-Stokes equations by strong and fast forcing to leverage inviscid approximations, which is a standard technique in the theory of controllability.

Gioacchino Antonelli, Marco Pozzetta

In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces. In this setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov--Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray. Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and we characterize the geometric rigidity at the level of the isoperimetric profile. Specializing our study to the $2$-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary. The proofs exploit a variational approach, and in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.

Giao Ky Duong, Nejla Nouaili, Hatem Zaag, Université Sorbonne Paris Nord, LAGA, CNRS(UMR7539), F-93430, Villetaneuse, France

In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a flat blow-up profile, already proved by Bricmont and Kupainen together with Herrero and Velázquez. Though our approach relies on the well-celebrated method, based on the reduction of the problem to a finite-dimensional one, then the use of a topological 'shooting method' to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first time where modulation is used with this kind of profiles. We do hope that this simplifies the argument.

Juan-Carlos Felipe-Navarro

Abstract In this work we prove the uniqueness of solutions to the nonlocal linear equation $$L \varphi - c(x)\varphi = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>φ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>c</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> , where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $$L u = f(u)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> when the nonlinearity is of Allen–Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli–Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.

Davit Harutyunyan, Narek Hovsepyan

This work is concerned with the study of the extreme rays of the convex cone of $3\times 3$ quasiconvex quadratic forms (denoted by ${\cal C}_3$). We characterize quadratic forms $f\in {\cal C}_3,$ the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. We prove that in the case when the determinant of the acoustic tensor of a form $f\in {\cal C}_3$ is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of ${\cal C}_3;$ when the determinant is a perfect square, then the form is either an extreme ray of ${\cal C}_3$ or polyconvex; and finally, when the determinant is identically zero, then the form $f$ must be polyconvex. The zero determinant case plays an important role in the proofs of the other two cases. We also make a conjecture on the extreme rays of ${\cal C}_3,$ and discuss about weak and strong etremals of ${\cal C}_d$ for $d\geq 3.$ where it turns out that several properties of ${\cal C}_3$ do not hold for ${\cal C}_d$ for $d&gt;3,$ and thus case $d=3$ is special. These results recover all previously known results (to our best knowledge) on examples of extreme points of ${\cal C}_3$ that were proved to be such. Our results also improve the ones proven by the first author and Milton [20].

Marco Doemeland

In this thesis we are interested in solving a class of quasilinear parabolic partial differential equations (PDEs) for closed differential forms which exhibit a special structure, namely a gradient flow structure, and thus bringing together two major areas of mathematical analysis, the geometric theory of differential forms and the theory of gradient flows. More precisely, for a bounded domain $ \Omega \subset \mathbb{R}^n $, $ n \ge 2 $, with smooth boundary and a time-dependent differential $ k $-form $ \omega (t) \colon \Omega \rightarrow \Lambda ^k ( \mathbb{R}^n ) $ we consider the gradient flow equation $\partial _t \omega \,=\, -\mathop{}\!\mathrm{d} \Big( \nabla c^ \ast \Big[ \mathop{}\!\mathrm{d}^{\ast} \big( \nabla _ \xi F( \, \cdot \, , \omega )\big) \Big] \Big) $ und $ \mathop{}\!\mathrm{d} \omega \,=\, 0 $. Here, $ c \colon \Lambda ^ {k-1}( \mathbb{R}^n ) \rightarrow [0, \infty )$, called the dissipation potential, is a convex function with Legendre-Fenchel dual $ c^ \ast $, whereas $ \nabla _ \xi F $ denotes the derivative of the energy density $ F \colon \Omega \times \Lambda ^k( \mathbb{R}^n ) \rightarrow \mathbb{R} $ with respect to its second argument. This class of PDEs was suggested by Yann Brenier in 2014 as a general framework for dissipative equations and contains for example the $ p $-Hodge Laplace heat equation for closed differential forms $ \partial _t \omega = - \mathop{}\!\mathrm{d} ( | \mathop{}\!\mathrm{d}^{\ast} \omega | ^ {p-2} \, \mathop{}\!\mathrm{d}^{\ast} \omega ) $. The problem of finding weak solutions of the gradient flow equation is challenging not only because of its nonlinearity, but also because of its vectorial character, i.e. it is a system of scalar PDEs. The gradient flow structure appears in the form of a so-called Energy Dissipation Inequality (EDI). Although the latter is equivalent to the gradient flow equation only on a formal level, it nevertheless plays an essential role in establishing the proof of the existence of weak solutions for the PDE. In order the prove the existence of solutions of the corresponding EDI, we use a so-called minimizing movement scheme. This is a time-discrete approximation scheme in which each time-step consists of solving a variational problem. The variational problem involves the perturbation of the energy functional with the so-called dissipation functional which is defined using the dissipation potential $ c $. This dissipation functional is closely related to the norms of the duals of homogeneous Sobolev spaces for differential forms, i.e. negative homogeneous Sobolev spaces, which are introduced here. Hence, these spaces define the natural functional analytic setting for the problems addressed in this thesis. To the best of our knowledge, this concept of negative homogeneous Sobolev spaces for differential forms is new. In the limit where the time discretization parameter, used to define the perturbed energy functional, tends to zero the approximation scheme weakly converges to some limit. Since the EDI has well-suited lower semicontinuity properties with respect to the weak convergence, the limit is indeed a solution of the EDI. The limiting process also benefits from compensated compactness methods such as the Sobolev-Poincaré inequality in combination with a Minty-Browder-type argument. With some extra effort we can also prove a reversed EDI for the limit. As the main result of this thesis we conclude from this the existence of a weak solution of the gradient flow equation. In the second part of the thesis we ask for additional properties of the weak solutions of the gradient flow equation such as uniqueness, a semigroup property of the time evolution, an exponential formula as well as error estimates. These problems are very difficult to solve. Because the concept of the EDI is too weak for these questions, we invoke the stronger concept of the Evolution Variational Inequality (EVI). The latter is formally equivalent to the gradient flow equation as well. However, it is only available in the case $ c( \xi ) = \frac{1}{2}| \xi | ^2 $ in which the dissipation functional becomes, up to a scalar multiple, the negative homogeneous Sobolev norm for the case of Hilbert spaces. As the main results for the second part we prove uniqueness of the limit found before in a class of admissible solutions of the EVI, a contraction property and a semigroup property of the time evolution as well as an exponential formula together with an error estimate. The proof of the exponential formula and the error estimate is given by using two different approaches.