Mathematics

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form ?�Δu+V(x)u = ( I α ?�|u | p )|u | p?? u+λ|u | q?? u,u??H 1 ( R N ), where λ>0,N??,α??0,N) . The potential V is a continuous function and I α denotes the standard Riesz potential. Assume also that 1<q<2, 2 α <p< 2 ??α where 2 α =(N+α)/N , 2 α =(N+α)/(N??) . Our main contribution is to consider a specific condition on the parameter λ>0 taking into account the nonlinear Rayleigh quotient. More precisely, there exists λ n >0 such that our main problem admits at least two positive solutions for each λ??0, λ n ] . In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter λ n >0 is optimal in some sense which allow us to apply the Nehari method.

We consider the fractional porous medium flow introduced by Caffarelli-Vazquez and obtain local in time existence, uniqueness, and blow-up criterion for smooth solutions. The proof is based on establishing a commutator estimate involving fractional Laplacian operators.

We study the existence of classical solutions to a broad class of local, first order, forward-backward Extended Mean Field Games systems, that includes standard Mean Field Games, Mean Field Games with congestion, and mean field type control problems. We work with a strictly monotone cost that may be fully coupled with the Hamiltonian, which is assumed to have superlinear growth. Following previous work on the standard first order Mean Field Games system, we prove the existence of smooth solutions under a coercivity condition that ensures a positive density of players, assuming a strict form of the uniqueness condition for the system. Our work relies on transforming the problem into a partial differential equation with oblique boundary conditions, which is elliptic precisely under the uniqueness condition.

We present a classification of strict limits of planar BV homeomorphisms. The authors and S. Hencl showed in a previous work \cite{CHKR} that such mappings allow for cavitations and fractures singularities but fulfill a suitable generalization of the INV condition. As pointed out by J. Ball \cite{B}, these features are physically expected by limit configurations of elastic deformations. In the present work we develop a suitable generalization of the \emph{no-crossing} condition introduced by De Philippis and Pratelli in \cite{PP} to describe weak limits of planar Sobolev homeomorphisms that we call \emph{BV no-crossing} condition, and we show that a planar mapping satisfies this property if and only if it can be approximated strictly by homeomorphisms of bounded variations.

Elliptic Problems with Dirichlet boundary conditions have been wisely investigated. In this paper, we generalize the results obtained in arXiv:1909.11950 to the case of p -Laplcace equation with Robin boundary conditions. The point-wise comparison, obtained in arXiv:1909.11950 only in the planar case, returns in any dimension when p is sufficiently small.

Starting from the local-in-time classical solution to the compressible Euler system with impermeable boundary condition in half-space, by employing the coupled weak viscous layers (governed by linearized compressible Prandtl equations with Robin boundary condition) and linear kinetic boundary layers, and the analytical tools in \cite{Guo-Jang-Jiang-2010-CPAM} and some new boundary estimates both for Prandtl and Knudsen layers, we proved the local-in-time existence of Hilbert expansion type classical solutions to the scaled Boltzmann equation with Maxwell reflection boundary condition with accommodation coefficient α ε =O( ε ??) when the Knudsen number ε small enough. As a consequence, this justifies the corresponding case of formal analysis in Sone's books \cite{Sone-2002book, Sone-2007-Book}. This also extends the results in \cite{GHW-2020} from specular to Maxwell reflection boundary condition. Both of this paper and \cite{GHW-2020} can be viewed as generalizations of Caflisch's classic work \cite{Caflish-1980-CPAM} to the cases with boundary.

In this paper, we study a nonlinear interaction problem between compressible viscous fluids and plates. For this problem, we introduce relative entropy and relative energy inequality for the finite energy weak solutions (FEWS). First, we prove that for all FEWS, the relative energy inequality is satisfied and that the structure displacement enjoys improved regularity by utilizing the dissipation effects of the fluid onto the structure. Then, we show that all FEWS enjoy the weak-strong uniqueness property, thus extending the classical result for compressible Navier-Stokes system to this fluid-structure interaction problem.

In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form ?u+f(u,?�u)=0 . By studying a well chosen fixed point problem defined around the numerical solution, these techniques make it possible to prove the existence of a solution in an explicit (and usually small) neighborhood the numerical solution. In this work, we develop a similar approach for a broader class of systems, including nonlinear diffusion terms of the form ?Φ(u) . In particular, this enables us to obtain new results about steady states of a cross-diffusion system from population dynamics: the (non-triangular) SKT model. We also revisit the idea of automatic differentiation in the context of computer-assisted proof, and propose an alternative approach based on differential-algebraic equations.

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system set on Ω? R 3 , for a smooth bounded domain Ω of R 3 , with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to 0 while the distribution function concentrates towards a Dirac mass in velocity centered at 0 , with an exponential rate. The proof, which follows the methods introduced in [Han-Kwan - Moussa - Moyano, arXiv:1902.03864v2], requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviors for the kinetic density, from total absorption to no absorption at all.

Let L=?+V be Schr{ö}dinger operator with a non-negative potential V on a complete Riemannian manifold M . We prove that the conical square functional associated with L is bounded on L p under different assumptions. This functional is defined by G L (f)(x)= ( ????0 ??B(x, t 1/2 ) |??e ?�tL f(y) | 2 +V| e ?�tL f(y) | 2 dtdy Vol(y, t 1/2 ) ) 1/2 . For p?�[2,+?? we show that it is sufficient to assume that the manifold has the volume doubling property whereas for p??1,2) we need extra assumptions of L p ??L 2 of diagonal estimates for { t ????e ?�tL ,t??} and { t ??V ??????e ?�tL ,t??} .Given a bounded holomorphic function F on some angular sector, we introduce the generalized conical vertical square functional G F L (f)(x)= ( ????0 ??B(x, t 1/2 ) |?�F(tL)f(y) | 2 +V|F(tL)f(y) | 2 dtdy Vol(y, t 1/2 ) ) 1/2 and prove its boundedness on L p if F has sufficient decay at zero and infinity. We also consider conical square functions associated with the Poisson semigroup, lower bounds, and make a link with the Riesz transform.