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Determining Projections and Functionals for Weak Solutions of the Navier-Stokes Equations  [PDF]
Michael Holst,Edriss Titi
Mathematics , 2010,
Abstract: In this paper we prove that an operator which projects weak solutions of the two- or three-dimensional Navier-Stokes equations onto a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies a single appoximation inequality. We then apply this result to show that the long-time behavior of weak solutions to the Navier-Stokes equations, in both two- and three-dimensions, is determined by the long-time behavior of a finite set of bounded linear functionals. These functionals are constructed by local surface averages of solutions over certain simplex volume elements, and are therefore well-defined for weak solutions. Moreover, these functionals define a projection operator which satisfies the necessary approximation inequality for our theory. We use the general theory to establish lower bounds on the simplex diameters in both two- and three-dimensions. Furthermore, in the three dimensional case we make a connection between their diameters and the Kolmogoroff dissipation small scale in turbulent flows.
Existence, uniqueness and stability of solutions of generalized Tikhonov-Phillips functionals  [PDF]
Gisela L. Mazzieri,Ruben D. Spies,Karina G. Temperini
Mathematics , 2011,
Abstract: The Tikhonov-Phillips method is widely used for regularizing ill-posed inverse problems mainly due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the corresponding optimization problems has originated a variety other methods which can be considered as "variants" of the traditional Tikhonov-Phillips method of order zero. Such is the case for instance of the Tikhonov-Phillips method of order one, the total variation regularization method, etc. In this article we find sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals which guarantee existence and uniqueness and stability of the minimizers. The particular cases in which the penalizers are given by the bounded variation norm, by powers of seminorms and by linear combinations of powers of seminorms associated to closed operators, are studied. Several examples are presented and a few results on image restoration are shown.
On Existence and Uniqueness of the Weak Solution of a Generalized Boussinesq Equation with Press and  [PDF]
Gol Kim,Bao-Zhu Guo
Mathematics , 2012,
Abstract: In this paper, a generalized Boussinesq equation that couples the mass and heat flows in a viscous incompressible uid is considered. The kinematic viscosity and the heat conductivity are assumed to be dependent on the temperature. The boundary condition on the velocity of fluid is non-standard where the dynamical pressure is given on some part of the boundary, and the temperature of fluid is represented in a mixed boundary condition. The existence of the weak solution is proved by the Galerkin approximation scheme, and the uniqueness is also obtained under the condition on the weak solution that is somehow like the restriction on the Reynold number and the Raleigh number in hydrodynamics.
Stochastic power law fluids: Existence and uniqueness of weak solutions  [PDF]
Yutaka Terasawa,Nobuo Yoshida
Mathematics , 2010, DOI: 10.1214/10-AAP741
Abstract: We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force. We investigate the existence and the uniqueness of weak solutions to this SPDE.
On the existence and uniqueness of weak solutions for a vorticity seeding model  [PDF]
L. C. Berselli,M. Romito
Mathematics , 2004,
Abstract: In this paper we study the Navier-Stokes equations with a Navier-type boundary condition that has been proposed as an alternative to common near wall models. The boundary condition we study, involving a linear relation between the tangential part of the velocity and the tangential part of the Cauchy stress-vector, is related to the vorticity seeding model introduced in the computational approach to turbulent flows. The presence of a point-wise non vanishing normal flux may be considered as a tool to avoid the use of phenomenological near wall models, in the boundary layer region. Furthermore, the analysis of the problem is suggested by recent advances in the study of Large Eddy Simulation. In the two dimensional case we prove existence and uniqueness of weak solutions, by using rather elementary tools, hopefully understandable also by applied people working on turbulent flows. The asymptotic behaviour of the solution, with respect to the averaging radius $\delta,$ is also studied. In particular, we prove convergence of the solutions toward the corresponding solutions of the Navier-Stokes equations with the usual no-slip boundary conditions, as the small parameter $\delta$ goes to zero.
Existence and uniqueness of weak and entropy solutions for homogeneous Neumann boundary-value problems involving variable exponents
Bernard K. Bonzi,Ismael Nyanquini,Stanislas Ouaro
Electronic Journal of Differential Equations , 2012,
Abstract: In this article we study the nonlinear homogeneous Neumann boundary-value problem $$displaylines{ b(u)-hbox{div} a(x, abla u)=fquad hbox{in } Omegacr a(x, abla u).eta=0 quadhbox{on }partial Omega, }$$ where $Omega$ is a smooth bounded open domain in $mathbb{R}^{N}$, $N geq 3$ and $eta$ the outer unit normal vector on $partialOmega$. We prove the existence and uniqueness of a weak solution for $f in L^{infty}(Omega)$ and the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.
Fractional porous media equations: existence and uniqueness of weak solutions with measure data  [PDF]
Gabriele Grillo,Matteo Muratori,Fabio Punzo
Mathematics , 2013,
Abstract: We prove existence and uniqueness of solutions to a class of porous media equations driven by the fractional Laplacian when the initial data are positive finite Radon measures on the Euclidean space. For given solutions without a prescribed initial condition, the problem of existence and uniqueness of the initial trace is also addressed. By the same methods we can also treat weighted fractional porous media equations, with a weight that can be singular at the origin, and must have a sufficiently slow decay at infinity (power-like). In particular, we show that the Barenblatt-type solutions exist and are unique. Such a result has a crucial role in [24], where the asymptotic behavior of solutions is investigated. Our uniqueness result solves a problem left open, even in the non-weighted case, in [42]
Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation  [PDF]
Geng Chen,Robin Ming Chen,Yue Liu
Mathematics , 2015,
Abstract: The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the H\"older continuous energy conservative solutions for the Cauchy problem of the Novikov equation.
Existence and uniqueness of weak solutions to viscous primitive equations for certain class of discontinuous initial data  [PDF]
Jinkai Li,Edriss S. Titi
Mathematics , 2015,
Abstract: We establish some conditional uniqueness of weak solutions to the viscous primitive equations, and as an application, we prove the global existence and uniqueness of weak solutions, with the initial data taken as small $L^\infty$ perturbations of functions in the space $X=\left\{v\in (L^6(\Omega))^2|\partial_zv\in (L^2(\Omega))^2\right\}$; in particular, the initial data are allowed to be discontinuous. Our result generalizes in a uniform way the result on the uniqueness of weak solutions with continuous initial data and that of the so-called $z$-weak solutions.
Global Existence and Uniqueness of Weak Solutions of 3-D Euler Equations with Helical Symmetry in the Absence of Vorticity Stretching  [PDF]
Boris Ettinger,Edriss S. Titi
Mathematics , 2008,
Abstract: We prove uniqueness and existence of the weak solutions of Euler equations with helical symmetry, with initial vorticity in $L^{\infty}$ under "no vorticity stretching" geometric constraint. Our article follows the argument of the seminal work of Yudovich. We adjust the argument to resolve the difficulties which are specific to the helical symmetry.
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