# Recent Advances in Partial Differential Equations and Applications

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*Vicenţiu D. Rădulescu; Adélia Sequeira; Vsevolod A. Solonnikov*

This volume contains the proceedings of the
International Conference on Recent Advances in PDEs and Applications,
in honor of Hugo Beirão da Veiga's 70th birthday, held from
February 17–21, 2014, in Levico Terme, Italy.

The conference brought together leading experts and researchers in
nonlinear partial differential equations to promote research and to
stimulate interactions among the participants. The workshop program
testified to the wide-ranging influence of Hugo Beirão da Veiga on
the field of partial differential equations, in particular those
related to fluid dynamics.

In his own work, da Veiga has been a seminal influence in many
important areas: Navier-Stokes equations, Stokes systems,
non-Newtonian fluids, Euler equations, regularity of solutions,
perturbation theory, vorticity phenomena, and nonlinear potential
theory, as well as various degenerate or singular models in
mathematical physics. This same breadth is reflected in the
mathematical papers included in this volume.

# Table of Contents

## Recent Advances in Partial Differential Equations and Applications

- Cover Cover11
- Title page v6
- Contents vii8
- Preface ix10
- Tributes to Hugo Beirão da Veiga 112
- Analyticity of the semi-group generated by the Stokes operator with Navier-type boundary conditions on 𝐿^{𝑝}-spaces 2334
- Some results on systems for quantum fluids 4152
- Remarks on the inviscid limit for the compressible flows 5566
- A generalization of Gauss’ divergence theorem 6980
- Weak solutions to the Navier-Stokes equations constructed by semi-discretization are suitable 8596
- Existence theory for generalized Newtonian fluids 99110
- The spectral drop problem 111122
- On the vanishing theorems for the discretely self-similar solutions to the Hall equations 137148
- A high regularity result of solutions to a modified 𝑝-Navier-Stokes system 151162
- General properties of the Helmholtz decomposition in spaces of 𝐿^{𝑞}-type 163174
- Conditional regularity of very weak solutions to the Navier-Stokes-Fourier system 179190
- Possible effect of noise on stretching mechanism 201212
- On the plane steady-state flow of a shear-thinning liquid past an obstacle in the singular case 211222
- Sectorial Hamiltonians without zero resonance in one dimension 225236
- Vortex stretching and anisotropic diffusion in the 3D Navier-Stokes equations 239250
- On 𝐿^{𝑞} estimates of planar flows up to the boundary 253264
- Non equilibrium diffusion limit in a barotropic radiative flow 265276
- Decomposition of the homogeneous space 𝑊^{1,2} with respect to the Dirichlet form ⟨∇𝑢,∇𝑣⟩ and applications 279290
- 1. Introduction 279290
- 2. Notations 280291
- 3. Decomposition of the homogeneous space 𝑊^{1,2} 280291
- 4. Variational formulation of the Stokes boundary value problem 282293
- 5. Lower and upper bounds to the change of the Dirichlet seminorm by Yosida approximation 282293
- 6. Bounds to transport-diffusion splitting schemes 283294
- 7. Lower and upper bounds to the change of vorticity by transition from slip- to no-slip fluid flow 285296
- References 286297

- Convection in ternary porous layers with depth-dependent permeability and viscosity 289300
- 1. Introduction 289300
- 2. Preliminaries 290301
- 3. Main link between the unknown fields 292303
- 4. Linear stability 294305
- 5. Stationary and overstable convection 296307
- 6. Nonlinear stability 297308
- 7. Global non linear stability via symmetries and skew-symmetries hidden in (2.6) 299310
- 8. Applications 300311
- References 302313

- On a variational inequality for incompressible non-Newtonian thick flows 305316
- On inhomogeneous 𝐩-Navier–Stokes systems 317328
- On the global well-posedness of some free boundary problem for a compressible barotropic viscous fluid flow 341352
- On a free boundary problem of magnetohydrodynamics for a viscous incompressible fluid not subjected to capillary forces 357368
- Relative entropy and contraction for extremal shocks of conservation laws up to a shift 385396
- Back Cover Back Cover1418