This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density $\rho$ and velocity field $u$ satisfy ${\parallel \nabla \rho \parallel }_{L^{\infty }(0,T;W^{1,q})}+{\parallel u\parallel }_{L^s(0,T;L_{\omega }^r)}<\infty$ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r\le 1$, $3<r\le \infty ,$ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here $L_{\omega }^r$ denotes the weak $L^r$ space.

We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result...

We consider a quasistatic contact problem for an electro-viscoelastic body. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We present a weak formulation for the model and establish existence and uniqueness results. The proofs are based on classical results for elliptic variational...

We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter ε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution...

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described...

The paper is dedicated to the global well-posedness of the barotropic compressible Navier-Stokes-Poisson system in the whole space ${ℝ}^N$ with N ≥ 3. The global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces. The initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations. The proof relies on a uniform estimate for a mixed hyperbolic/parabolic linear system with a convection term.

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They...

In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.

In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.

The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $-\Delta u-{\Lambda }_N\frac{u}{{\left|x\right|}^2}={\left|\nabla u+\frac{N-2}{2}\frac{u}{{\left|x\right|}^2}x\right|}^2{\left|x\right|}^{\left(N-2\right)/2}+\lambda f\left(x\right)$ in $\Omega$, $u=0$ on $\partial \Omega$, ${\Lambda }_N={(\left(N-2\right)/2)}^2$. This problem is a particular case of problem (2). Notice that $\left(N-2\right)/2$ is optimal as coefficient and exponent on the right hand side.

We study a nonlinear elliptic system with resonance part and nonlinear boundary conditions on an unbounded domain. Our approach is variational and is based on the well known Landesman-Laser type conditions.

Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized...

The incompressible three-dimensional Navier-Stokes equations are considered. A new regularity criterion for weak solutions is established in terms of the pressure gradient.

In Albano-Cannarsa [1] the authors proved that, under some conditions, the singularities of the semiconcave viscosity solutions of the Hamilton-Jacobi equation propagate along generalized characteristics. In this note we will provide a simple proof of this interesting result.

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial }_{t}u+\frac{.}{\dot{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with bounded divergence....

In this paper we propose a solution of the Lambertian shape-from-shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solution. The power of our approach is twofolds: (1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Moreover, it allows to characterize the viscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal.29 (1992) 867–884],...