[Paper Review] Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density

Jinkai Li|arXiv (Cornell University)|May 5, 2016

Navier-Stokes equation solutions13 references3 citations

TL;DR

This paper establishes the local existence and uniqueness of strong solutions to the 3D incompressible Navier-Stokes equations with nonnegative initial density, including vacuum, under weaker regularity assumptions than previous works. It removes the need for a compatibility condition and reduces the required regularity of the initial velocity, relying instead on momentum continuity at the initial time.

ABSTRACT

In this paper, we consider the initial-boundary value problem to the nonhomogeneous incompressible Navier-Stokes equations. Local strong solutions are established, for any initial data $(ρ_0, u_0)\in (W^{1,γ} \cap L^\infty) imes H_{0,σ}^1$, with $γ>1$, and if $γ\geq2$, then the strong solution is unique. The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The assumption on the initial data is weaker than the previous widely used one that $(ρ_0, u_0)\in (H^1 \cap L^\infty ) imes(H_{0,σ}^1 \cap H^2)$, and no compatibility condition is required.

Motivation & Objective

• To establish local strong solutions for the nonhomogeneous incompressible Navier-Stokes equations with initial density allowed to be zero (vacuum).
• To weaken the standard assumptions on initial data, particularly removing the need for a compatibility condition.
• To reduce the required regularity of the initial velocity from $H^2$ to $H^1$, while preserving uniqueness.
• To show that continuity of the momentum $\rho u$ at $t=0$ is sufficient, rather than continuity of velocity $u$ itself.

Proposed method

• Uses a Galerkin approximation scheme with regularized initial data to construct approximate solutions $ (\rho_n, u_n) $.
• Employs energy estimates and $ L^p $-type inequalities to control the evolution of $ \rho u $, $ \nabla u $, and $ \rho $.
• Applies the Gronwall-type inequality to control the difference between two solutions and prove uniqueness.
• Uses the continuity of $ \rho u $ at $ t=0 $ as the initial condition, rather than $ u $, due to potential discontinuity in velocity at vacuum.
• Relies on Sobolev, Poincaré, and Hölder inequalities to bound nonlinear terms in the momentum and density equations.
• Establishes $ L^2 $-continuity of $ \rho u $ at $ t=0 $ via limit arguments on approximating sequences, proving the initial condition holds in the momentum sense.

Experimental results

Research questions

• RQ1 Can strong solutions exist for the nonhomogeneous Navier-Stokes equations with initial vacuum, under weaker regularity assumptions?
• RQ2 Is it possible to remove the compatibility condition $ \Delta u_0 - \nabla p_0 = \sqrt{\rho_0} g $ while preserving uniqueness?
• RQ3 Can the initial velocity regularity be reduced from $ H^2 \cap H^1_0 $ to $ H^1_0 $ without losing uniqueness?
• RQ4 Does continuity of the momentum $ \rho u $ at $ t=0 $ suffice as an initial condition when $ u $ may be discontinuous at vacuum?
• RQ5 Can uniqueness be proven under $ \gamma \geq 2 $ with only $ (\rho_0, u_0) \in (W^{1,\gamma} \cap L^\infty) \times H^1_{0,\sigma} $, no compatibility?

Key findings

• Local strong solutions exist for initial data $ (\rho_0, u_0) \in (W^{1,\gamma} \cap L^\infty) \times H^1_{0,\sigma} $ with $ \gamma > 1 $, allowing initial vacuum.
• Uniqueness is established when $ \gamma \geq 2 $, under the same initial data regularity.
• The compatibility condition required in prior works is completely removed.
• The initial condition is imposed on the momentum $ \rho u $, not on $ u $, and continuity of $ \rho u $ at $ t=0 $ is proven via approximation.
• The regularity of the initial velocity is reduced from $ H^2 \cap H^1_0 $ to $ H^1_0 $, while preserving uniqueness.
• The solution satisfies $ \rho u \in C([0,T]; L^2) $, and $ \sqrt{t} u, \sqrt{t} \partial_t u \in L^\infty(0,T; H^2) \cap L^2(0,T; H^1) $, indicating improved time-regularity near $ t=0 $.