Compressible Navier-Stokes-Riesz System Analysis

• Compressible Navier-Stokes-Riesz system is a nonlinear PDE framework that models viscous compressible fluids under fractional attractive forces and physical vacuum conditions.
• It incorporates nonlocal Riesz interactions and weighted Sobolev norms to prove global existence, regularity, and algebraic decay towards compact steady states.
• Analytic techniques leverage weighted energy methods and Hardy-type inequalities to handle vacuum degeneracy and ensure stability and convergence under precise parameter constraints.

The compressible Navier-Stokes-Riesz system is a nonlinear, nonlocal partial differential equation (PDE) framework modeling viscous compressible fluids subject to an attractive Riesz potential, in regimes where both vacuum boundaries and fractional interaction effects are prominent. The system generalizes classical compressible Navier-Stokes equations by incorporating attractive forces with fractional order, leading to challenging analysis especially in the presence of physical vacuum and time-evolving free boundaries. Recent work establishes sharp global existence, regularity, and Lyapunov-type stability results for strong solutions in one dimension, with physical parameters and initial data precisely constrained to ensure asymptotic convergence toward compactly-supported steady states (Carrillo et al., 3 Jan 2026).

1. System Formulation and Nonlocal Potential

The primary model consists of the compressible Navier-Stokes equations with a nonlocal Riesz interaction term, posed on a moving interval $0 \leq x \leq a(t)$, where vacuum is present at $x = a(t)$. The equations in Eulerian form, after normalization and for effective viscosity $4\mu/3+\lambda=1$, are: $$\begin{align*} \partial_t \rho + \partial_x (\rho u) &= 0, \ \partial_t (\rho u) + \partial_x (\rho u^2) + \partial_x p(\rho) &= \partial_x(\partial_x u) - \rho \, \partial_x (\Phi*\rho), \end{align*}$$ where $p(\rho) = \rho^\gamma$ and the attractive Riesz potential kernel is $\Phi(x) = |x|^{2s-1}/(2s-1)$, with $0 < s < 1/2$. The nonlocal force $\rho \, \partial_x (\Phi*\rho)$ introduces fractional interaction via convolution: $$(\Phi*\rho)(x) = \int_\mathbb{R}\Phi(x-y)\rho(y) dy$$.

Physical-vacuum free boundary conditions enforce

$$\rho(t, a(t)) = 0,\quad u(t, 0) = 0,\quad \partial_x u(t, a(t)) = 0,$$

and the free boundary evolves by $\dot{a}(t) = u(t,a(t))$.

2. Lagrangian Reformulation and Steady States

Translating to Lagrangian coordinates on the fixed interval $x \in [-R, R]$ leverages the particle mapping $n(t,x)$, solving $n_t = u(t, n(t,x))$, with $n(0,x) = n_0(x)$ and boundary condition $n(\pm R) = \pm a(t)$. Use the notation $d(t,x) = \rho(t, n(t,x))$, $v(t,x) = u(t, n(t,x))$, and steady density profile $p(x)$ satisfying

$d(t,x)\, n_x(t,x) = p(x).$

The momentum equation in Lagrangian variables is: $$\partial_t(d v) + \partial_x \left[ d^{-1} (p(x))^2 \right] - \partial_x \Psi(t,x) = 0,$$ with the nonlocal term

$$\Psi(t,x) = \partial_x p(x) + \int_{-R}^{R} W'(n(t,x)-n(t,y))[p(y) n_x(t,x) - p(x) n_x(t,y)] dy,$$

where $W'(z) = -|z|^{2s-1}$. The stationary profile $p(x)>0$ solves $p''(x) + \partial_x (p\, \partial_x \Phi*p) = 0$; it is compactly supported in $[-R,R]$, $p(x) \sim (1-|x|)$ near $|x|=R$, reflecting the “physical vacuum” regime.

3. Parameter Regimes and Initial Data Constraints

Rigorous analysis is achieved under constraints for the fractional parameter $s$ and adiabatic exponent $\gamma$:

• $0 < s < 1/2$,
• $2(1-s) < \gamma < 1+2s/3$,
• For uniqueness and certain regularity, $3/8 < s < 1/2$ is imposed.

Initial conditions $n_0(x)$, $v_0(x)$ must correspond to small perturbations of the steady state (the identity map and zero velocity, respectively), with compatibility in weighted Sobolev norms. In particular,

$$n_0(\pm R) = \pm a_0,\quad n_{0x}(\pm R)=1,\quad v_0(0) = 0,\quad n_0(0) = 0,$$

and match steady sound speed at the boundary in terms of derivatives and weighted $H^1$ norms.

4. Existence, Regularity, and Stability Results

The main results demonstrate:

• Global-in-time strong solution existence: For initial data satisfying $\|n_{0x}-1\|_{H^1_w} + \|v_{0x}\|_{L^2_w} \ll 1$, there exists a unique strong solution over $[0,\infty)\times[-R,R]$.
• Uniform Jacobian bounds: $0<C_1\leq n_x(t,x)\leq C_2<\infty$ for all $(t,x)$, guaranteeing invertibility.
• Regularity: $n-x$ is in $C([0,\infty); H^2_w)$, $n_t\in C([0,\infty); H^1_w)$, and higher derivatives are locally square integrable in weighted spaces.
• Lyapunov-type energy bound: The energy $$E(t) = \|n_x(t)-1\|_{H^1_w}^2 + \|n_t(t)\|_{H^1_w}^2$$ obeys $E(t)\leq C E(0)$ for all $t$.
• Algebraic decay: For $i=0,1$, $$\|\partial_x^i(n_x-1)(t)\|_{L^2} + \|\partial_x^i n_t(t)\|_{L^2}\leq C E(0) (1+t)^{-2}$$, and second-order derivatives also decay at rate $(1+t)^{-2}$.
• Convergence to steady state: Solutions in weighted Sobolev norms converge with polynomial rate to the steady profile $p(x)$.

5. Analytic Techniques and Handling Degeneracy

The analysis hinges on weighted energy methods, exploiting a Lyapunov-type functional adapted to the vacuum boundary and nonlocal interaction. The perturbation $w=n-x$ leads to a damped wave equation with nonlocal source, incorporated in the energy

$$E_0(t) = \int_{-R}^R p(x)\left[w_t^2 + (1+2\partial_x w)^{\gamma-1}(\partial_x w)^2\right] dx + \int_{-R}^R\int_{-R}^R p(x)p(y)\mathcal{R}[w] dx\,dy,$$

where $\mathcal{R}[w]$ captures Riesz interaction at quadratic order. The direct computation yields $dE_0/dt\leq 0$, guaranteeing monotonic energy dissipation.

Near the vacuum boundary ($x=\pm R$), where $p(x)\sim(R-|x|)$, analysis in weighted Sobolev spaces $H^k_w$ leverages boundary-sensitive norms and Hardy-type inequalities to control degeneracy. Nonlocal terms are addressed using symmetry, Young’s inequality, and a decomposition into near-field and far-field components. In particular, the operator

$L w = \int \frac{w(x) - w(y)}{|x-y|^{1+2s}} dy$

admits a coercivity estimate: $$\int w\, Lw\, dx \geq 2(1-s)\int\int |w(x)-w(y)|^2 p(x)p(y) |x-y|^{-(3-2s)} dx\,dy,$$ which ensures pressure dominance when $\gamma > 2(1-s)$.

Continuation relies on a key bootstrap yielding $n_x(t,x)$ close to $1$, together with a finite-difference scheme and global energy bounds.

6. Weighted Sobolev Norms and Energy-Dissipation Structure

Analysis heavily employs weighted Sobolev norms with weights determined by the steady density profile: $$\|f\|^2_{L^2_w} := \int_{-R}^R p(x) f(x)^2 dx, \quad \|f\|^2_{H^1_w} := \|f\|^2_{L^2_w} + \|\partial_x f\|^2_{L^2_w}.$$ Final energy-dissipation inequalities quantify temporal decay: $$\frac{d}{dt} E_k(t) + C D_k(t) \leq 0, \quad E_k(t) \leq E_k(0) (1+t)^{-m},$$ where $E_k(t)$ involves weighted norms of time/spatial derivatives of $w$ and $m>0$ depends on $(\gamma, s)$.

7. Implications and Mathematical Significance

The results reveal the robust stability of compactly-supported steady states for compressible Navier-Stokes-Riesz systems in the presence of vacuum, provided explicit parameter and data constraints. The analytic framework unifies diffusion, nonlocal attraction, and physical vacuum effects in a single model, demonstrating that the interplay of pressure and nonlocal attraction—parameterized by $\gamma$ and $s$—governs solution regularity and long-time asymptotics. The methodology, particularly weighted energy techniques and nonlocal commutator estimates, is expected to generalize to multi-dimensional and more complex nonlocal systems with vacuum boundaries.

The analysis articulates the minimal conditions for global existence, uniqueness, regularity, and convergence, presenting a comprehensive template for future research on compressible systems with fractional or nonlocal interactions (Carrillo et al., 3 Jan 2026).