Journal de Mathematiques Pures et Appliquees最新文献

We analyze a system of PDEs governing the interaction between two compressible mutually noninteracting fluids and a shell of Koiter type encompassing a time dependent 3D domain filled by the fluids. The dynamics of the fluids is modeled by a system resembling compressible Navier-Stokes equations with a physically realistic pressure depending on densities of both the fluids. The shell possesses a non-linear, non-convex Koiter energy. Considering that the densities are comparable initially we prove the existence of a weak solution until the degeneracy of the energy or the self-intersection of the structure occurs for two cases. In the first case the adiabatic exponents are assumed to satisfy $\max \{\gamma ,\beta \}>2$, $\min \{\gamma ,\beta \}>0$, and the structure involved is assumed to be non-dissipative. For the second case we assume the critical case $\max \{\gamma ,\beta \}\ge 2$ and $\min \{\gamma ,\beta \}>0$ and the dissipativity of the structure. The result is achieved in several steps involving, extension of the physical domain, penalization of the interface condition, artificial regularization of the shell energy and the pressure, the almost compactness argument, added structural dissipation and suitable limit passages depending on uniform estimates.

We prove new optimal $C^{1,\alpha }$ regularity results for obstacle problems involving evolutionary p-Laplace type operators in the degenerate regime $p>2$. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward p-paraboloids, up to the critical exponent, $\alpha \le 2/(p-2)$, and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or $C^{1,\alpha }$ intrinsic interpolative geometries.

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to $O\left({\nu }^{1/4}\right)$ order terms in $L^{\infty }$ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces.

In addition, we also prove that monotonic boundary layer profiles, which are stable when $\nu =0$, are nonlinearly unstable when $\nu >0$, provided ν is small enough, up to $O\left({\nu }^{1/4}\right)$ terms in $L^{\infty }$ norm.

Huisken's problem asks whether there is an elastic flow of closed planar curves that is initially contained in the upper half-plane but ‘migrates’ to the lower half-plane at a positive time. Here we consider variants of Huisken's problem for open curves under the natural boundary condition, and construct various migrating elastic flows both analytically and numerically.

In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is well known that, without any monotonicity conditions, mean field games typically contain multiple mean field equilibria and the wellposedness of their corresponding master equations fails. In this paper, a partial order for the set of probability measure flows is proposed to compare different mean field equilibria. The minimal and maximal mean field equilibria under this partial order are constructed and satisfy the flow property. The corresponding value functions, however, are in general discontinuous. We thus introduce a notion of weak-viscosity solutions for the master equation and verify that the value functions are indeed weak-viscosity solutions. Moreover, a comparison principle for weak-viscosity semi-solutions is established and thus these two value functions serve as the minimal and maximal weak-viscosity solutions in appropriate sense. In particular, when these two value functions coincide, the value function becomes the unique weak-viscosity solution to the master equation. The novelties of the work persist even when restricted to the standard mean field games.

In this work, we study a particular system of coagulation equations characterized by two values, namely volume v and surface area a. Compared to the standard one-dimensional models, this model incorporates additional information about the geometry of the particles. We describe the coagulation process as a combination between collision and fusion of particles. We prove that we are able to recover the standard one-dimensional coagulation model when fusion happens quickly and that we are able to recover an equation in which particles interact and form a ramified like system in time when fusion happens slowly.

This paper proposes a geometrisation of $N$-manifolds of degree n as n-fold vector bundles equipped with a (signed) $S_n$-symmetry. More precisely, it proves an equivalence between the categories of $\left[n\right]$-manifolds and the category of (signed) symmetric n-fold vector bundles, by finding that symmetric n-fold vector bundle cocycles and $\left[n\right]$-manifold cocycles are identical.

This extends the already known equivalences of [1]-manifolds with vector bundles, and of [2]-manifolds with involutive double vector bundles, where the involution is understood as an $S_2$-action.

We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schrödinger equation of the form$-\Delta u+\lambda u=g\left(u\right),u\in H^1\left(R^N\right),N\ge 1.$ Our approach permits to handle in a unified way nonlinearities $g\left(s\right)$ which are either mass subcritical, mass critical or mass supercritical. Among its main ingredients is the study of the asymptotic behaviors of the positive solutions as $\lambda \to 0^{+}$ or $\lambda \to +\infty$ and the existence of an unbounded continuum of solutions in $(0,+\infty )\times H^1\left(R^N\right)$.

The primitive equations for geophysical flows are studied under the influence of stochastic wind driven boundary conditions modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic $L_t^q$-$H_z^{-1,p}{L}_{xy}^p$-setting. This solution is constructed in critical spaces.

In this paper, we study a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the stability of its separable solutions.