Here, we extend the weak KAM and Aubry-Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton-Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtai...

We study the existence of weak solutions to a Cahn–Hilliard–Darcy system coupled with a convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy’s law. The system of equations arises from a mixture model for tumour growth accounting for transport mechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, the existence of g...

We establish rigorously the existence of a three-parameter family of self-similar, globally bounded, and continuous weak solutions in two space dimensions to the compressible Euler equations with axisymmetry for γ-law polytropic gases with 1 ≤ γ < 2. The initial data of these solutions have constant densities and outward-swirling velocities. We use the axisymmetry and self-similarity assumption...

In this paper we analyze the three-dimensional Peterlin viscoelastic model. By means of a mixed Galerkin and semigroup approach prove existence weak solutions. Further combining parabolic regularity with relative energy method derive conditional weak-strong uniqueness result.

In this paper, we study the time periodic problem to a three-dimensional chemotaxis-Stokes model with porous medium diffusion Δ n m and inhomogeneous mixed boundary conditions. By using double-level approximation method some iterative techniques, obtain existence time-space uniform boundedness of weak solutions for any &gt; 1. Moreover, improve regularity [Formula: see text] show that obtained ...

We study the unsteady incompressible Navier-Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter type. This leads to coupled system PDEs where moving part boundary is an unknown problem. The known existence theory for weak solutions extended models. introduce a-priori estimates that reveal higher regularity displacement beyond energy estimates. These are e...

The model studied concerns a simple first-order hyperbolic system. The solutions in which one is most interested have discontinuities which persist for all time, and therefore need to be interpreted as weak solutions. We demonstrate existence and uniqueness for such weak solutions, identifying a canonical ‘exact’ solution which is everywhere defined. The direct method used is guided by the theo...

This paper is concerned with Sobolev weak solution of Hamilton-Jacobi-Bellman (HJB) equation. This equation is derived from the dynamic programming principle in the study of the stochastic optimal control problem. Adopting Doob-Meyer decomposition theorem as one of main tool, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. For the ...