Multiple integrals in the calculus of variations and nonlinear elliptic systems

We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution . In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

A remark on Liouville-type theorems for the stationary Navier–Stokes equations in three space dimensions

We consider the Navier-Stokes equation in a domain with a rough boundary. The roughness is modeled by a small amplitude and small wavelength oscillation, with typical scale ${\varepsilon \ll 1}$. For periodic oscillation, it is well-known that the best homogenized (that is regular in ${\varepsilon}$) boundary condition is of Navier type. Such result still holds for random stationary irregularities, as shown recently by the first author [5, 15]. We study here arbitrary irregularity patterns.

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Relevance of the Slip Condition for Fluid Flows Near an Irregular Boundary

Lp-theory for Stokes and Navier–Stokes equations with Navier boundary condition

http://ncmm.karlin.mff.cuni.cz/preprints/06341153340BFN06_01.pdf

On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries

On Liouville type theorems for the steady Navier–Stokes equations in R3

We study partial regularity of weak solutions of the three-dimensional (3D) valued nonstationary Hall magnetohydrodynamics equations on $\mathbb{R}^2$. In particular, we prove the existence of a weak solution whose set of possible singularities has the space-time Hausdorff dimension at most two.

On Partial Regularity for the 3D Nonstationary Hall Magnetohydrodynamics Equations on the Plane

In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$. In the first theorem we improve logarithmically the well-known $L^{\frac92} (\Bbb R^3)$ result. In the second theorem we present a sufficient condition for the trivially of the solution($v=0$) in terms of the head pressure, $Q=\frac12 |v|^2 +p$. The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee $v=0$.

On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$

We prove Liouville type theorems for the self-similar solutions to the Navier–Stokes equations. One of our results generalizes the previous ones by Necas–Ružicka–Sverak and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier–Stokes equations with the profile belonging to ${L^{p, \infty} (\mathbb{R}^3)}$ with ${p > \frac{3}{2}}$.

Joerg Wolf

Papers

On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries

On Liouville type theorems for the steady Navier–Stokes equations in R3

On Partial Regularity for the 3D Nonstationary Hall Magnetohydrodynamics Equations on the Plane

On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$

On the Liouville Type Theorems for Self-Similar Solutions to the Navier–Stokes Equations