Venue

The workshop will take place at the University of Minho, Auditorium of the Congregados building (information here).

Schedule

The abstract part deals with $d\times d$ tensors $A(x)$ that are symmetric, positive semi-definite, and whose row-wise divergence vanishes identically, or is controlled in the space of bounded measures. We establish the integrability of $(\det A)^{\frac1{d-1}}$, with sharp inequalities. The latter quantity failing to be concave over ${\small\bf Sym}_d^+$, unlike $(\det A)^{\frac1d}$, this result does not follow from Jensen's inequality.
The proof is a consequence of a duality between this structure and the elliptic Monge-Ampère equation; it involves Brenier's Theorem on optimal transport of measures.
Compensated Integrability has interesting geometrical consequences, like a new proof of the isoperimetric inequality. Extreme divergence-free tensors are solutions of Minkowski's Problem about convex bodies.
When applied to gas dynamics, Compensated Integrability yields new a priori estimates of space-time integrals, when the total mass and energy are finite. We cover all scales, from micro- to macroscopic ones: Euler and Boltzman equations, molecular dynamics, etc. The newly discovered integrability concerns either the quantity $\rho^{\frac1n}p$, where $\rho$ is the mass density, $p$ the pressure and $n$ the space dimension, or $$\left(\int_{\mathbb R^n}^{\otimes(n+1)}f(\xi_0)\cdots f(\xi_n)\left[{\rm vol}\left({\rm Simplex}(\xi_0,\ldots,\xi_n)\right)\right]^2d\xi_0\cdots d\xi_n\right)^{\frac1n}$$ where $f(\xi)=f(t,x,\xi)$ is the particle density in kinetic models. In molecular dynamics, we estimate the number of non-weak and non-grazing collisions.

[1] D. Serre. Divergence-free positive symmetric tensors and fluid dynamics. Annales de l'IHP, analyse non linéaire, 35 (2018), pp 1209-1234.
[2] D. Serre. Compensated integrability. Applications to the Vlasov-Poisson equation and other models of mathematical physics. Journal de Mathématiques Pures et Appliquées, accepted.
[3] D. Serre. Estimating the number and the strength of collisions in molecular dynamics. Preprint.

Processes involving multicomponent diffusion and chemical reactions appear in many applications in fluid mechanics and chemistry. The diffusive behavior of the species is well described by the equations introduced by Maxwell and Stefan, which provide a more general and appropriate framework than the standard Fickian approach. In this talk, we consider a chemically reactive mixture described in the frame of the Boltzmann equation and study the reaction-diffusion limit of the kinetic system of equations. Under certain assumptions, we formally derive a reaction-diffusion system of Maxwell-Stefan type. The emphasis is on the contributions resulting from the chemical reaction and the case of a reactive mixture of polyatomic gases is also analysed.

Lyapunov functions are used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano's method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.

Except for linear evolution PDEs for which the least square method may be applied,it seems hopeless to reduce nonlinear initial value problems to a convex optimization problem. We show that this is possible for all systems of conservation laws with a convex entropy and also for some of their singular limits, such as the Euler equations of incompressible fluids. The resulting convex problems look like matrix-valued generalizations of optimal transport or mean-field games problems. The sharpness of the method is discussed in the elementary case of the Burgers equation without viscosity.

Shadow wave is a picewise constant functions with respect to the time variable parametrized by a small parameter. We say that it solve a conservation or balance law system if its substitution into the system gives zero as the parameter tends to zero. The main use of these objects are systems having unbounded solutions. We are addressing the problem of proper choice if such solutions and the problem of their interactions with the other waves. The main goal is to find a solution to initial data problem tracking these interactions.

The porous medium model is an interacting particle system which belongs to the class of Kinetically constrained lattice gases (KCLG). In this talk, we are interested in studying the hydrodynamic limit of this model in contact with slow reservoirs, which guarantees that the evolution of the density of particles of this model is described by the weak solution of the corresponding hydrodynamic equation, namely, the porous medium equation with Dirichlet, Neumann and Robin boundary conditions, depending on the parameter that rules the slowness of the reservoirs.

We develop a regularity theory for fully nonlinear integro-differential equations with kernels deforming in space like sections of a convex solution of a Monge-Ampère equation. We prove an ABP estimate and a Harnack inequality and derive Hölder and $C^{1,\alpha}$ regularity results for solutions.
(joint work with L.A. Caffarelli and J.M. Urbano)

Since Y. Brenier's works in the 90's it is now understood that the Euler equation for inviscid fluids can be viewed as an incompressible version of Monge-Kantorovich optimal transport. On the other hand, a popular and numerically useful model within the optimal transport community is the so-called Schrödinger problem, which can be viewed as a stochastic entropic regularization of the Monge-Kantorovich problem, and Gamma-converges to the latter as the viscosity $\nu\to 0$. Recently, M. Arnaudon, AB. Cruzeiro, C. Léonard & JC. Zambrini introduced the so-called Brödinger problem.
In this talk I will show that, in the limit of small noise $\nu\to 0$, the Brödinger problem also Gamma-converges towards Brenier's problem. In other words, Euler's incompressible inviscid motion of fluids can be recovered in a variational setting from the viscous Brödinger counterpart (loosely related to Navier-Stokes flows), with a clear underlying stochastic interpretation. As a byproduct we obtain new results for the time-convexity of the entropy along several dynamical interpolations of measures and also recover earlier results of R. McCann and C. Léonard.
This is a joint work with Aymeric Baradat.

We are interested here in questions related to the regularity of solutions of elliptic problems with Dirichlet or Neumann boundary condition (see [1]). For the last 20 years, lots of work has been concerned with questions when $\Omega$ is a Lipschitz domain.
We give here some complements for the case of the Laplacian (see [3]), the Bilaplacian ([2],[6]) and the operator $\mathrm{div}\, (A \nabla)$ (see [5]), when ${\bf A}$ is a matrix or a function, and we extend this study to obtain other regularity results for domains having an adequate regularity.
Using the duality method, we will then revisit the work of Lions-Magenes [4], concerning the so-called ery weak solutions, when the data are less regular. Thanks to the interpolation theory, it permits us to extend the classes of solutions and then to obtain new results of regularity.
This is a joint work with Mohand Moussaoui (Ecole Normale Supérieure de Kouba, Alger) and Huy Hoang Nguyen (Univ. Federal do Rio de Janeiro, Brazil).

[1] C. Amrouche, M. Moussaoui, H.H. Nguyen. Laplace equation in smooth or non smooth domains. Work in Progress.
[2] B.E.J. Dahlberg, C.E. Kenig, J. Pipher, G.C. Verchota. Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier, 47, no. 5, 1425-1461, (1997).
[3] D. Jerison, C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161-219, (1995).
[4] J.L. Lions, E. Magenes. Problèmes aux limites non-homogènes et applications, Vol. 1, Dunod, Paris, (1969). [5] J. Necas}. \emph{ Direct methods in the theory of elliptic equations}. Springer Monographs in Mathematics. Springer, Heidelberg, (2012).
[6] G.C. Verchota . The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194 no. 2, 217-279, (2005).

In this talk we will present and discuss some properties of the weak solutions to the following generalization of the well known parabolic p-Laplacian $$\partial_t(u^{q})- \nabla \cdot \left(|\nabla u|^{p-2} \nabla u \right) =0 ,\ 0 < q,\ p < 1$$ modeling the turbulent filtration of a non-Newtonian fluid through a porous medium.
The double nonlinearity brings extra difficulties and we will see how a proper choice of the adopted geometry, combined with intrinsic scaling, an alternative argument and some simple but crucial estimates to the trick for the study of boundedness and continuity of the local weak solutions.

We study the criterions (sufficient conditions) for the local in time existence of a strong solution to the Navier-Stokes equations with a given initial velocity $u_0\in L^3(\Omega)$. Robinson, Rodrigo and Sadowski proved this criterion for solutions of the Navier-Stokes equations on the whole space. Kučera, Píšová and Vacková proved this criterion for solution of the Navier-Stokes equations with Navier type boundary conditions. Kučera, Píšová and Skalák proved this criterion for solution of the Navier-Stokes equations with Navier's boundary conditions.

We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev-Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Our results extend and complement those of [F. Weissler, ADE 2001], particularly because we are working in the larger setting of Sobolev-weak-$L^p$ spaces and considering non-local terms. The two nonlinearities of power-type and the generality of the non-local terms allow us to cover in a unified way a large number of dispersive equations and systems.

In this talk we will consider the Schrödinger system arizing in nonlinear optics $$ \left\{\begin{array}{lllll} \displaystyle iu_t+\Delta u-u+\Big(\frac{1}{9}|u|^2+2|w|^2\Big)u+\frac{1}{3}\overline{u}^2w=0,\\ i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0. \end{array}\right. $$ Here, $(x,t)\in \mathbb R^n\times\mathbb R$, $1\leq n\leq 3$ and $\sigma,\mu>0$.
This system can be put in the more general form $$U_t=JH'(U),$$ where $U=(u,w)$, $J$ is skew-adjoint and the Hamiltonian $H$ is an $H^1$-functional. We will discuss the existence of ground states, their semi-trivial/fully non-trivial nature and (in)stability.
This is a joint work with Ademir Pastor - Universidade Estadual de Campinas (Unicamp), Brazil.

In this talk, we will start by reviewing some basic notions of General Relativity, in particular the structure of black holes in spherical symmetry, leading to the formulation of the celebrated strong cosmic censorship conjecture, from a PDE perspective. We will then describe the characteristic initial value problem for the study of the Einstein-Maxwell-scalar field system inside a black hole, as a model for studying this conjecture, recalling some of the known results in this context.
We will finish by presenting recent work, where improvement is achieved by considering the presence of a cosmological constant and imposing more realistic initial data along the event horizon of the black hole, satisfying a Price law. The conclusions focus on the occurrence of mass inflation or the extendability of the corresponding maximal globally hyperbolic development.
This is joint work with João L. Costa, Pedro Girão and José Natário.

In this talk I will present existence results for localized vector solutions of the cubic nonlinear Helmholtz system \begin{align*} -\Delta u - \mu u &= u^3 + buv^2 \qquad\text{in }\mathbb{R}^N, \\ -\Delta v - \nu u &= v^3 + bvu^2 \,\qquad\text{in }\mathbb{R}^N \end{align*} for given $\mu,\nu>0$ and a coupling parameter $b\in\mathbb{R}$. Our results are obtained using a dual variational approach and bifurcation theory. The talk is based on joint work with R. Mandel. It is supported by the German Research Foundation (DFG) through CRC 1173 "Wave phenomena: analysis and numerics''.

We consider the modified Korteweg-de Vries equation over $\mathbb{R}$ $$u_t + u_{xxx}=\pm(u^3)_x.$$ This equation arises, for example, in the theory of water waves and vortex filaments in fluid dynamics. A particular class of solutions to (mKdV) are those which do not change under scaling transformations, the so-called \textit{self-similar} solutions. Self-similar solutions blow-up when $t\to 0$ and determine the asymptotic behaviour of the evolution problem at $t=+\infty$.
The known local well-posedness results for the (mKdV) fail when one considers critical spaces, where the norm is scaling-invariant. This also means that self-similar solutions lie outside of the scope of these results. Consequently, the dynamics of (mKdV) around self-similar solutions are also unknown.
In this talk, we will show existence and uniqueness of solutions to the (mKdV) lying on a critical space which includes both regular and self-similar solutions. Afterwards, we present several results regarding global existence, asymptotic behaviour at $t=+\infty$ and blow-up phenomena at $t=0$.
This is joint work with Raphaël Côte and Luis Vega.

This talk presents new existence results on the solution of an evolutionary Monge-Kantorovich type equation in an open domain with Dirichlet boundary conditions. The solution is given by a couple $(\lambda, u)$, where $\lambda$ is the transport density and u is the potential solving an evolution variational inequality with a time dependent nonlocal gradient constraint. Using a penalisation of the transport flux we extend to the evolution case the existence of Lagrange multipliers, which includes the case of a "nonlocal sand pile" problem.
This is a joint work with Assis Azevedo and Lisa Santos.

In this talk we discuss a recent work of the speaker that obtains an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(\alpha,d)>0$ such that $$\|I_\alpha \nabla u \|_{L^{d/(d-\alpha),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C \|\nabla u\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)}$$ for all $u \in \dot{W}^{1,1}(\mathbb{R}^d)$. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$.

We consider a free boundary optimization problem for the fractional Laplacian with a volume constraint and a lower temperature bound. We prove the existence and the optimal regularity of solutions. Moreover, two natural free boundaries arise in our problem. So, we provide not only geometric properties of solutions, but also of the corresponding exterior and interior free boundaries.
This is a joint work with V. Nersesyan (Université Paris-Saclay) and R. Teymurazyan (University of Coimbra).

In this talk I will present several PDEs driven by fractional operators with different types of boundary conditions. These PDEs can be obtained from a scaling limit of different microscopic stochastic dynamics and they are ruling the space-time evolution of the conserved quantity of the system.