Complete classification of gradient blow-up and recovery of boundary condition for the viscous Hamilton-Jacobi equation

Noriko Mizoguchi; Philippe Souplet

Home/Archive/arXiv

arXiv · 2110.12934

Complete classification of gradient blow-up and recovery of boundary condition for the viscous Hamilton-Jacobi equation

Noriko Mizoguchi·Philippe Souplet

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the Cauchy-Dirichlet pbm for superquadratic viscous Hamilton-Jacobi eq. We give a complete classification, namely rates and space-time profiles, in 1d case when viscosity sol. undergo gradient blow-up (GBU) or recovery of boundary condition (RBC) at any time when such phenomenon occurs. These results can be modified in radial domains in general dimensions. Previously, upper and lower estimates of GBU or RBC rates were available only in special case when basic comparison principle can be used. Even for type II BU in other PDEs, as far as we know, there has been no complete classification except [50], in which the argument relies on features peculiar to chemotaxis syst. Whereas there are many results on construction of special type II BU sol. of PDEs with investigation of (in-)stability of bubble, determination of (in-)stability of space-time profile for general sol. has not been done. In this paper, we determine whether space-time profile for each sol. is stable or unstable. A key in our proofs is to focus on algebraic structure with respect to vanishing intersections with singular steady state. In turn, GBU and RBC rates and profiles, as well as their (in-)stability, can be completely characterized by the number of vanishing intersections. We construct special sol. in bounded and unbounded intervals in both GBU and RBC cases, based on methods from [29], and then apply braid group theory to get upper and lower estimates of the rates. After that, we rule out oscillation of the rates, which leads us to the complete space-time profile. In the process, careful construction of special sol. with specific behaviors in intermediate and outer regions, far from bubble and the RBC point, plays essential role. The application of such techniques to viscosity sol. is completely new.

Explore related subjects

math.AP

Keep this discovery

BibTeX RIS

Noriko Mizoguchi, Philippe Souplet. 2021-10-25. Complete classification of gradient blow-up and recovery of boundary condition for the viscous Hamilton-Jacobi equation. https://arxiv.org/abs/2110.12934

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points

This paper is devoted to the well-posedness of classical Prandtl equations in a finite order Sobolev space. For a initial data with degenerate critical points and general outflow, we obtain the local-in-time existence and uniqueness of the solution to the Prandtl equations in a Sobolev space, by introducing a new iteration scheme and linear cancelation. This result shows that Oleinik's monotonicity condition is not a necessary condition for the Prandtl equations to be well-posed in Sobolev spaces and provides evidence to demonstrate that zero shear stress does not necessarily lead to boundary layer separation in two-dimensional unsteady boundary layers.

math.AP↗

arXiv2026

Global existence and time decay for a bipolar Euler-Poisson system with one pressureless and undamped fluid

We study the Cauchy problem for a three-dimensional bipolar Euler--Poisson system in which one fluid is pressureless and undamped, while the other is subject to momentum relaxation. For sufficiently small smooth perturbations of a constant equilibrium, we prove the global existence and uniqueness of smooth solutions under an irrotationality assumption on the initial velocity of the pressureless fluid, together with algebraic time-decay estimates. The main difficulty is that the velocity of the pressureless fluid is dissipated only indirectly through the Poisson coupling, and this mechanism degenerates strongly at high frequencies, leading to a regularity-loss structure. We overcome this difficulty by combining refined Green-function estimates, a low--middle--high frequency decomposition, and high-order nonlinear energy estimates adapted to the asymmetric regularity hierarchy. The result establishes a global small-data theory for this asymmetric regime, in which pressure and damping are simultaneously absent from the same fluid.

math.AP↗

arXiv2026

Boundary layer of 2D Chemotaxis Navier-Stokes equations with logarithmic Sensitivity. II. viscous vanishing limit

This is the second part of a two-part work concerning boundary layer solutions to the coupled Chemotaxis-Navier-Stokes system in the two-dimensional half-space. In the present work, we address the convergence of boundary layer solutions to singular chemotaxis-fluid equations under slip boundary conditions with respect to the chemical diffusion-viscosity parameter $\varepsilon$ in the two-dimensional half-plane. More precisely, we show that the boundary layer for $\varepsilon>0$ (viscous convection coefficient) converges to the superposition of the outer layer (solution with $\varepsilon=0$) and the inner layer as $\varepsilon\rightarrow0$. The outer and inner profiles are explicitly derived as in the first part\cite{WWZ}. Furthermore, the well-posedness results of the coupled Chemotaxis-Navier-Stokes system in conormal Sobolev spaces will be presented in Appendix. They answer the question mentioned in the first part of the two-part work. This study could help the understanding of the chemotactic movement of aerobic bacteria to the water-air surface observed experimentally in fluids, and enrich the theoretical results of boundary layer in chemotactic fluid models.

math.AP↗