We prove the local in time existence of solutions for an aggregation equation in Besov spaces. The Fourier localization technique and Littlewood-Paley theory are the main tools used in the proof.

This paper introduces two novel nonlinear anisotropic Picone identities with variable exponents that expand upon the traditional identity used for the ordinary Laplace equation. Additionally, the research explores potential applications of these findings in anisotropic Sobolev spaces featuring variable exponents.

We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.

We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u({\eta }_1,t),\cdots ,u({\eta }_q,t)$ with $0\le {\eta }_1<{\eta }_2<\cdots <{\eta }_q<1.$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $\left({\mathrm{P}}_q\right)$ of (P) in which the nonlinear term contains the sum $S_q\left[u^2\right]\left(t\right)=q^{-1}{\sum }_{i=1}^q{u}^2(\frac{(i-1)}{q},t)$. Under suitable conditions, we prove that the solution of $\left({\mathrm{P}}_q\right)$ converges to the solution of the corresponding...

The Dirichlet boundary value problem for systems of elliptic partial differential equations at resonance is studied. The existence of a unique generalized solution is proved using a new min-max principle and a global inversion theorem.

We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. Suitable boundary conditions on the traction at the inflow/outflow part of boundary are given. As an advantage of this, the mean value of the pressure over the domain is no more a free parameter which would have to be prescribed otherwise. We prove the existence and uniqueness of weak solutions (the latter for small data) and discuss particular applications of the results.

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing...

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta –1}V\left(B^{\beta }\right)$. For the interaction parameter $\beta \in (0,2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N\to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0,3/5)$ in [28]. This allows, in the case $\beta \in (0,3/5)$, for the application of the Klainerman–Machedon uniqueness theorem...

We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i\theta }$ for all $\theta \in ℝ$. We investigate about the persistence of the decorrelation between the...

We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on Cm-smoothness of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional...

Cet exposé présente les résultats de l’article [3] au sujet des ondes progressives pour l’équation de Gross-Pitaevskii : la construction d’une branche d’ondes progressives non constantes d’énergie finie en dimensions deux et trois par un argument variationnel de minimisation sous contraintes, ainsi que la non-existence d’ondes progressives non constantes d’énergie petite en dimension trois.

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T4$ configurations. We then use this to construct weak solutions to the unstable interface problem (the...