Differentiable selections of multifunctions and their applications

In this paper, the Cauchy problem for nonstrictly hyperbolic systems of conservation laws modeling polymer flooding in enhanced oil recovery is studied, and a short proof of global existence of weak entropy solutions is obtained by using a new technique from the compensated compactness theorem coupled with the total variation estimate on one family of Riemann invariants. This work extends in some sense the previous works [T. Barkve, SIAM J. Appl. Math., 49 (1989), pp. 789--798; T. Johansen and R. Winther, SIAM J. Math. Anal., 19 (1988), pp. 541--566] about the global existence of weak solutions for the Riemann problem and by the author [J. Funct. Anal., 261 (2011), pp. 2797--2815] about the global weak solution of the Cauchy problem for an extended hyperbolic system of Keyfitz--Kranzer or Aw--Rascle type.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We study the Cauchy problem for scalar hyperbolic conservation laws with a flux that can have jump discontinuities. We introduce new concepts of entropy weak and measure-valued solution that are consistent with the standard ones if the flux is continuous. Having various definitions of solutions to the problem, we then answer the question what kind of properties the flux should possess in order to establish the existence and/or uniqueness of solution of a particular type. In any space dimension we establish the existence of measure-valued entropy solution for a flux having countable jump discontinuities. Under the additional assumption on the Hölder continuity of the flux at zero, we prove the uniqueness of entropy measure-valued solution, and as a consequence, we establish the existence and uniqueness of weak entropy solution. If we restrict ourselves to one spatial dimension, we prove the existence of weak solution to the problem where the flux has merely monotone jumps; in such a setting we do not require any continuity of the flux at zero.

We consider the stationary Navier-Stokes equations with nonhomogeneous boundary condition in a domain with several boundary components. If the boundary value satisfies only the necessary flux condition (GOC), Leray's inequality does not holds true in general and we cannot prove the existence of a solution. But for a 2-D domain which is symmetric with respect to a line and where the data is also symmetric, C. Amick showed the existence of solutions by reduction to absurdity; later H. Fujita proved Leray-Fujita's inequality and hence the existence of symmetric solutions. In this paper we give a new short proof of Leray-Fujita's inequality and hence a proof of the existence of weak solutions.

We consider a nonlinear integro-differential equation for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The equation can be written as a quasilinear Cauchy problem. For bounded reaction rates we prove global existence and uniqueness of classical solutions by means of evolution operator theory. We also prove global existence of weak solutions for unbounded reaction rates by a compactness argument.

We study the existence of global weak solutions for Cauchy-Dirichlet problem for evolutional p-Laplacian systems. The initial data is belonging to the Sobolev space W1,p(Ω, ℝn). We use a varitional (like) method to treat the existence of a weak solution, this method seems somehow simply than the Galerkin method.

A fourth-order partial differential model in this paper is called the thin-film equation in the field of fluid theory. The existence of the weak solution is obtained by solving two approximation problems. In order to perform the limit for small parameters in the approximation problems, the Galerkin method and the entropy functional method are both used. By means of some classical compactness results we can give the existence of nonnegative weak solutions. Finally, by redefining the entropy functional, the long-time exponential decay is given in the sense of L1-norm.

We show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosities. In analogy with the case of the compressible Navier-Stokes equations, the weak solutions satisfy the basic energy inequality, the Bresh-Desjardins entropy inequality and the Mellet-Vasseur estimate. These estimates play an important role in establishing the compactness of the vertical velocity of the approximating solutions, and therefore are essential to recover the vertical velocity in the weak solutions.

We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.

The main purpose of this article is to propose a reasonable definition for the stochastic integration (S.I.) of progressive processes w.r.t. semimartingales. This S.I. generalizes that of predictable processes w.r.t. semimartingales as well as the stochastic Stieltjes integration. This S.I. is proposed in §1. We give also in §1 an exponential formula for semimartingales using this S.I.. The rest of this paper consists of several remarks on the theory of stochastic integration which are mostly of pedagogical interest. In §2 we propose a new construction of the S.I. of predictable processes w.r.t. local martingales. A simple proof of the integration by parts formula is given in §3. Finally, we propose in §4 a short proof of Meyer’s theorem on compensated stochastic integrals of local martingales.AMS Subject Classification60H05

We show the existence of global weak solutions to a general class of kinetic models of homogeneous incompressible dilute polymers. The main new feature of the model is the presence of a general implicit constitutive equation relating the viscous part of the Cauchy stress and the symmetric part of the velocity gradient. We consider implicit relations that generate maximal monotone (possibly multivalued) graphs, and the corresponding rate of dissipation is characterized by the sum of a Young function and its conjugate depending on and , respectively. Such a framework is very general and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or Herschel–Bulkley type, and shear-rate dependent fluids with discontinuous viscosities as special cases. The appearance of and in all the assumptions characterizing the implicit relationship is fully symmetric. The elastic properties of the flow, characterizing the response of polymer macromolecules in the viscous solvent, are modeled by the elastic part of the Cauchy stress tensor, whose divergence appears on the right-hand side of the momentum equation, and which is defined by the Kramers expression involving the probability density function, associated with the random motion of the polymer molecules in the solvent. The probability density function satisfies a Fokker–Planck equation, which is nonlinearly coupled to the momentum equation. We establish long-time and large-data existence of weak solutions to such a system, completed by an initial condition and either a no-slip or Navier's slip boundary condition, by using properties of maximal monotone operators and Lipschitz approximations of Sobolev-space-valued Bochner functions via a weak compactness arguments based on the Div-Curl Lemma and Chacon's Biting Lemma. A key ingredient in the proof is the strong compactness in L 1 of the sequence of Galerkin approximations to the probability density function and of the associated sequence of approximations to the elastic part of the Cauchy stress tensor.

We aim at proving existence of weak solutions to the stationary compressible Navier-Stokes system coupled with the Allen-Cahn equation - NSAC system. The model is studied in a bounded three dimensional domain with slip boundary conditions for the momentum equations and the Neumann condition for the Allen-Cahn model. The main result establishes existence of weak solutions with bounded densities. The construction is possible assuming sufficiently large value of the heat capacity ratio $\gamma$ \hbox{$ (p \sim \varrho^\gamma$)}. As a corollary we obtain weak solutions for a less restrictive case losing pointwise boundedness of the density.

We consider a reaction–diffusion system of 2 × 2 equations modeling the spread of early tumor cells. The existence of weak solutions is ensured by a classical argument of Faedo–Galerkin method. Then, we present a numerical scheme for this model based on a finite volume method. We establish the existence of discrete solutions to this scheme, and we show that it converges to a weak solution. Finally, some numerical simulations are reported with pattern formation examples.

In the present paper we prove the existence of weak solutions $$ u:Q \to \mathbb{R}^{n}$$ to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder Q = Ω × (0,T), where $$ \Omega \subset \mathbb{R}^{n} $$ denotes an open set. For the power-low model with $$ q > 2\frac{{n + 1}}{{n + 2}}$$ we are able to construct a weak solution $$ u\in L^q(0, T; W_0^{1,q}(\Omega)^n)\cap C_w([0,T];L^2(\Omega)^n)$$ with ∇ · u = 0.