Pith. sign in

REVIEW 2 major objections 4 minor 43 references

On a free boundary problem of the Boltzmann equation

T0 review · 2 major / 4 minor · reviewed 2026-07-31 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims the first global well-posedness theory for a fully coupled Boltzmann equation with a moving piston, proving that small perturbations of the gas and piston decay exponentially to equilibrium.

desk verdict First rigorous global theory for a fully coupled Boltzmann piston, but the L∞ bootstrap has a load-bearing gap: Lemma 5.1 is unproved and Lemma 5.2 controls a boundary term by the interior norm. read the letter →

arxiv 2607.24053 v1 pith:ZKSLKS4J submitted 2026-07-27 math.AP math-phmath.MP

Shengchuang Chang , Renjun Duan , Shuangqian Liu This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35Q2035R3535B2035B4535B40
keywords Boltzmannequationgas-pistoninteractionfreeboundaryproblemglobalexistenceasymptoticstabilityconformaltransformationexponentialdecaykinetictheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies a one-dimensional piston problem in which a rarefied gas is confined between a fixed wall and a moving piston, with the piston driven by Newton's law under the gas drag. Because the gas domain moves with the piston and solutions have low regularity, the usual Lagrangian change of variables is unavailable. The authors introduce a conformal transformation that fixes the domain, then uncover a coupled quadratic energy structure that links the gas distribution to the piston position and velocity. Their main theorem states that, for small initial data satisfying a mass-compatibility condition, a unique nonnegative solution exists globally in time and converges exponentially to a global Maxwellian equilibrium. If correct, this is the first rigorous global result for a fully coupled kinetic free-boundary problem of piston type.

What carries the argument

The central object is the conformal transformation x1 = (1+X1)/(1+xw(t)), which maps the time-dependent physical interval onto a fixed interval and converts the free-boundary problem into a fixed-domain problem with coefficients depending on xw and vw. The load-bearing identity is a coupled quadratic energy: adding the piston energy to the L2 estimate of the gas fluctuation f produces exact cancellation of the boundary interaction terms, leaving a strictly positive dissipation coefficient for the piston velocity. Damping of the piston position is obtained not from Hooke's law alone but from a mass-flux test function xw(1+xw)x1v1√μ, using mass conservation to extract |xw|². The L∞ control is

What would settle it

A direct check of the stochastic-cycle measure (8.18) for the hard-sphere kernel: if the set of trajectories undergoing k=C T^{5/4} reflections does not have measure bounded by (1/2)^{C T^{5/4}}, the L∞ estimate (5.2) fails. Alternatively, a numerical simulation of the hard-sphere piston with small data satisfying the mass condition should show exponential decay of |xw(t)|, |vw(t)|, and ∥wf(t)∥∞; any algebraic or slower decay would contradict Theorem 2.1.

Watch

Extended reading notes

Core claim

Theorem 2.1 asserts that for initial data sufficiently small in a velocity-weighted L∞ norm, with finite weighted W^{1,p} spatial derivative and the mass compatibility condition ∫√μ f0 = -xw0/(1+xw0), there exists a unique nonnegative solution to the coupled Boltzmann–Newton system. The solution satisfies ∥wf(t)∥∞ + |xw(t)| + |vw(t)| ≤ C e^{-λt}(∥wf0∥∞ + |xw0| + |vw0|). The proof combines a conformal map fixing the moving interval, a coupled energy identity that produces dissipation both in the gas and in the piston dynamics, and an L∞–L2 framework with stochastic-cycle estimates to control the nonlinear collision terms.

Load-bearing premise

The argument rests on the stochastic-cycle estimate behind Lemma 8.2 and Proposition 5.1: that after finitely many wall reflections, the measure of surviving particle trajectories decays geometrically, and on the asserted but omitted proof of Lemma 5.1 that converts this into the velocity-weighted L∞ bound (5.2). If this cycle contraction fails, the L∞ bootstrap and the continuation argument do not close.

Editorial extensions

If this is right

  • If Theorem 2.1 is correct, global existence, uniqueness, nonlinear stability, and exponential convergence to equilibrium hold for the fully coupled piston problem near a global Maxwellian.
  • The damping of the piston position works even without a Hookean restoring force (κ=0), because the kinetic mass flux provides the missing dissipation.
  • The local-in-time weighted W^{1,p} estimates imply L^{1+δ} stability, which in turn gives uniqueness of the solution.
  • The structural ingredients—conformal fixing of the domain and coupled-energy cancellation—are expected to extend to other kinetic free-boundary and gas-structure interaction problems.
  • The authors state the results extend in a straightforward way to more general angular-cutoff collision kernels with hard or Maxwell-molecule potentials.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exponential decay proven here stands in sharp contrast to the algebraic decay found in collisionless and BGK-type numerical studies of pistons; a natural extrapolation is that in this bounded, mass-coupled, near-equilibrium setting, collisions destroy the long-time memory effect and restore exponential damping.
  • A testable extension is the κ=0 case with larger initial displacement: the theory predicts exponential relaxation at a rate independent of the spring constant, which could be compared against the algebraic t^{-3/2} decay seen in earlier numerical work on collisional pistons.
  • The conformal map and mass-flux cancellation may transfer to multi-dimensional slab geometries, where total mass and momentum conservation over the moving cell would play the role of the one-dimensional mass flux identity.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies a one-dimensional free-boundary problem for the Boltzmann equation coupled to a Newtonian piston, with diffuse reflection at a moving wall and a fixed wall. A nonlinear conformal transformation fixes the moving domain, and the perturbation around a global Maxwellian is written as f. The authors claim global existence, uniqueness, nonlinear stability, and exponential decay for small data satisfying a mass-compatibility condition (Theorem 2.1). The strategy combines a coupled L2–L∞ framework: an energy inequality for the pair (f, x_w, v_w), macroscopic dissipation estimates via dual test functions, a stochastic-cycle L∞ estimate, local weighted W^{1,p} estimates, and an L^{1+δ} stability argument for uniqueness.

Significance. If the proof is correct, this would be the first rigorous global well-posedness theory for a fully coupled Boltzmann free-boundary problem of piston type. The paper contains several valuable structural ideas: the conformal transformation, the coupled energy identity exposing a cancellation between kinetic boundary terms and piston dynamics, and an iterative construction of local solutions with nonnegativity. The a priori smallness hypothesis is a standard bootstrap assumption, and the mass compatibility condition is an initial-data constraint; I see no evidence of circularity. However, the global conclusion hinges on the velocity-weighted L∞ estimate of Proposition 5.1, and the proof of that estimate contains a load-bearing omitted lemma and an unaddressed compatibility issue in the stochastic-cycle estimates. These gaps are substantial but appear fixable in principle.

major comments (2)
  1. [§5, Lemma 5.2] Lemma 5.1 is stated with its proof omitted as “simpler than Lemma 8.1,” but it is load-bearing: the finite-time contraction (5.12) and the induction (5.26) in Proposition 5.1 rely on it. Lemma 8.1 is not an acceptable substitute because it concerns the iterative approximate sequence with νℓ, rℓ and v^ℓ_w, whereas Lemma 5.1 is for the actual solution. More seriously, in the final term (5.10), Lemma 5.2 bounds |h(t_k,x_1^k,v^{k-1})| by sup_s e^{ν0 s/2}‖h(s)‖∞, an interior norm. For a transport equation with diffuse reflection, an incoming boundary trace is not controlled by the interior L∞ norm unless one first applies the boundary condition at t_k and then estimates the resulting outgoing integral, e.g. with Lemma 8.2. The manuscript does not supply that step. Without a complete proof of Lemma 5.1, the finite-time estimate (5.12) and the exponential decay claim (2.19) are not verified.
  2. [§8, Lemma 8.2] Condition (8.19) requires sup_s max_ℓ |v^ℓ_w(s)| ≤ 1/(√(2π) k), with k = C_4 T_0^{5/4}. In the iteration one only has the a priori bound |v^ℓ_w| ≤ 2ε0, and no argument reconciles this with 1/(√(2π) k): T_0 is chosen in (5.12)/(8.18), and ε0 is chosen in Theorem 2.1, but the two choices are never made compatible. Since Lemma 5.2 invokes the estimates (8.20)–(8.21) that depend on (8.19), the finite-time L∞ contraction is not established under the hypotheses actually available. An explicit ordering of the smallness threshold ε0 and the time step T_0 (or a direct non-iterative analogue of (8.19)) is needed for the cycle bounds to close.
minor comments (4)
  1. [§3, Eq. (3.19)] The displayed constant identity after (3.19) is arithmetically inconsistent. From the preceding formulas the coefficient should be (25π−4)/(2√(2π)) rather than (8+π)/(2√(2π)) after subtracting 2√(2π). The coefficient is still positive, so this appears to be a typo, but it should be corrected because the positivity of the dissipation is used to close the L2 estimate.
  2. [§4, Lemma 4.1] Lemma 4.1 says the solution solves “(2.11) and (2.11)”; the second should be (2.12). The same typo appears in Lemmas 4.2 and 5.2/Proposition 5.1 statements.
  3. [§5, Lemma 5.1] The sentence “The proof of Lemma 5.1 is simpler than that of Lemma 8.1 later, and is therefore omitted for brevity” is not acceptable in a paper whose main theorem depends on this estimate. At minimum, the proof should be included in an appendix, and the differences from the iterative Lemma 8.1 should be spelled out.
  4. [§8, Lemma 8.2] The proof of Lemma 8.2 contains several notational inconsistencies, e.g. t_{\ell-(k-1)}^k versus t_{\ell-(k-1)}^{\ell} in (8.18), and the phrase “there exist as least” should be “there exist at least”. Please proofread the index bookkeeping carefully, since the cycle estimates are delicate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a self-contained analytical bootstrap with no fitted-input predictions and no load-bearing self-citation chain.

full rationale

I walked the claimed derivation chain: conformal rescaling to fix the domain, the reformulated Boltzmann–Newton system, the coupled L2/energy identities in Sections 3–4, the L∞ bootstrap in Section 5, the weighted W^{1,p} estimates in Section 6, the L^{1+δ} stability in Section 7, and the continuation/local-existence argument in Section 8. No step reduces by construction to its own input. The smallness condition (2.18) is an a priori hypothesis that is later closed by the continuation argument; it is not a fitted parameter and no quantity called a 'prediction' is generated from a subset of the same data. The mass compatibility condition in (2.18) is a constraint on the initial data used to identify the equilibrium mass; it is not an output of the theorem. The coupled energy structure is derived from the equations, and the dissipation of x_w is obtained through the identity (4.3), which follows from mass conservation (Lemma 4.2) rather than being assumed. The paper does not rely on self-citations: the cited results, such as [8], [21], [24], [25], are external prior works with non-overlapping authors with respect to the present paper, and they are used as technical tools (e.g., stochastic-cycle measure estimates and kinetic distance weights), not as the source of the central well-posedness claim. The only notable gap is the omitted proof of Lemma 5.1, which is a correctness risk (the claimed L∞ bootstrap is not fully demonstrated in the text), but an omitted or incomplete proof is not circularity: it does not make the derivation equivalent to its inputs by construction. Therefore the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard Boltzmann analysis tools, the hard-sphere and diffuse-reflection modeling assumptions, and the small-data/compatibility assumptions of Theorem 2.1. No empirical free parameters or invented physical entities are introduced; all constants are existential or tied to the model's physical inputs.

assumptions (6)
  • domain assumption Hard-sphere collision kernel B(v−u,ω)=|(v−u)·ω|
    Assumed throughout; extension to angular cutoff kernels is claimed but not proven (Section 1).
  • domain assumption Diffuse reflection boundary conditions (1.7)–(1.8) at both walls and equilibrium reservoir condition (1.4) to the right of the piston
    Defines the interaction model; the reservoir condition is central to the boundary dissipation structure and the xw-damping when κ=0.
  • domain assumption Mass normalization and smallness of initial data: ||wf0||∞+|xw0|+|vw0|≤ε0 with ∫√μ f0 dx = −xw0/(1+xw0)
    Initial data class in Theorem 2.1; the mass identity (4.3) is used to produce dissipation for the piston position in (3.23).
  • domain assumption Slab symmetry: one spatial dimension with velocity space R^3
    The entire theorem is 1D; multi-dimensional gas-structure interaction is explicitly left as future work.
  • domain assumption A priori smallness bound sup_t (||wf(t)||∞+|vw(t)|+|xw(t)|)≤2ε0
    Used in Propositions 3.1, 4.1, 5.1, 6.2; later closed by local existence and continuation in Section 8.
  • standard math Background Boltzmann estimates: coercivity of L=ν−K, kernel properties, L∞–L2 framework, stochastic cycle estimates from [8,21,24,25]
    Invoked throughout Sections 4–8; not reproved; assumed valid as published mathematical results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On a free boundary problem of the Boltzmann equation." pith.science (2026). https://pith.science/paper/ZKSLKS4J

@misc{pith2026260724053,
  author       = {Pith},
  title        = {Pith review of: On a free boundary problem of the Boltzmann equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKSLKS4J}},
  note         = {Machine review of arXiv:2607.24053}
}
read the original abstract

This paper studies a free boundary problem for the Boltzmann equation that models the interaction of rarefied gas with a moving wall---a classical piston problem in kinetic theory. The piston motion is governed by Newton's law under the drag force exerted by the gas, with or without an additional Hookean restoring force. A central challenge for such a problem is the strong coupling between the kinetic equation and the free moving boundary, for which the classical Lagrangian formulation is unavailable due to low-regularity of solutions. Our approach relies on two new ingredients: a conformal transformation is introduced to reduce the moving domain to a fixed domain, and a coupled energy structure linking the kinetic distribution and free boundary variables is uncovered to reveal intrinsic dissipation and cancellation mechanisms at the interface. These structural observations lead to a global nonlinear theory for the fully coupled system. As a result, we establish the global existence, uniqueness, nonlinear stability, and exponential convergence to equilibrium of solutions near a global Maxwellian. This provides the first rigorous global well-posedness theory for a fully coupled Boltzmann free boundary problem of piston type.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references

  1. [1]

    K. Aoki, G. Cavallaro, C. Marchioro and M. Pulvirenti, On the motion of a body in thermal equilibrium immersed in a perfect gas.M2AN Math. Model. Numer. Anal.42(2008), no. 2, 263–275

    2008

  2. [2]

    Aoki and F

    K. Aoki and F. Golse, On the speed of approach to equilibrium for a collisionless gas.Kinet. Relat. Models4(2011), no. 1, 87–107

    2011

  3. [3]

    K. Aoki, T. Tsuji and G. Cavallaro, Approach to steady motion of a plate moving in a free-molecular gas under a constant external force.Phys. Rev. E80(2009), 016309

    2009

  4. [4]

    Beale, Large-time regularity of viscous surface waves.Arch

    J. Beale, Large-time regularity of viscous surface waves.Arch. Rational Mech. Anal.84(1984), no. 4, 307–352

    1984

  5. [5]

    Bodn´ ar, G

    T. Bodn´ ar, G. Galdi and ˇS. Neˇ casov´ a, eds.,Fluid-Structure Interaction and Biomedical Applications, Adv. Math. Fluid Mech., Birkh¨ auser/Springer, 2014

    2014

  6. [6]

    Butt` a, G

    P. Butt` a, G. Cavallaro, and C. Marchioro,Mathematical Models of Viscous Friction.Lecture Notes in Mathematics, 2135. Springer, Cham, 2015. xiv+134 pp

    2015

  7. [7]

    ˇCani´ c, Moving boundary problems.Bull

    S. ˇCani´ c, Moving boundary problems.Bull. Amer. Math. Soc. (N.S.)58(2021), 79–106

    2021

  8. [8]

    Y. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains.Arch. Ration. Mech. Anal.233(2019), no. 3, 1027–1130

    2019

Show all 43 references
  1. [9]

    Caprino, G

    S. Caprino, G. Cavallaro, and C. Marchioro, On a microscopic model of viscous friction.Math. Models Methods Appl. Sci. 17(2007), no. 9, 1369–1403

    2007

  2. [10]

    Caprino, C

    S. Caprino, C. Marchioro and M. Pulvirenti, Approach to equilibrium in a microscopic model of friction.Comm. Math. Phys.264(2006), no. 1, 167–189

    2006

  3. [11]

    Cavallaro, On the motion of a convex body interacting with a perfect gas in the mean-field approximation.Rend

    G. Cavallaro, On the motion of a convex body interacting with a perfect gas in the mean-field approximation.Rend. Mat. Appl.(7)27(2007), no. 2, 123–145

    2007

  4. [12]

    Cavallaro and C

    G. Cavallaro and C. Marchioro, On the motion of an elastic body in a free gas.Rep. Math. Phys.69(2012), no. 2, 251–264

    2012

  5. [13]

    Cercignani,The Boltzmann Equation and Its Applications

    C. Cercignani,The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988

    1988

  6. [14]

    Chen and C

    H. Chen and C. Kim, Regularity of stationary Boltzmann equation in convex domains.Arch. Ration. Mech. Anal.244 (2022), no. 3, 1099–1222

    2022

  7. [15]

    Chen and W

    X. Chen and W. Strauss, Approach to equilibrium of a body colliding specularly and diffusely with a sea of particles.Arch. Ration. Mech. Anal.211(2014), no. 3, 879–910

    2014

  8. [16]

    Chen and W

    X. Chen and W. Strauss, Velocity reversal criterion of a body immersed in a sea of particles.Comm. Math. Phys.338 (2015), no. 1, 139–168

    2015

  9. [17]

    Chen and W

    X. Chen and W. Strauss, Convergence to equilibrium of a body moving in a kinetic sea.SIAM J. Math. Anal.47(2015), no. 6, 4630–4651

    2015

  10. [18]

    Coutand and S

    D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension.J. Amer. Math. Soc.20(2007), no. 3, 829–930

    2007

  11. [19]

    Coutand and S

    D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional com- pressible Euler equations in physical vacuum.Arch. Ration. Mech. Anal.206(2012), no. 2, 515–616

    2012

  12. [20]

    Dechrist´ e and L

    G. Dechrist´ e and L. Mieussens, A Cartesian cut cell method for rarefied flow simulations around moving obstacles.J. Comput. Phys.314(2016), 465–488

    2016

  13. [21]

    Esposito, Y

    R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law.Commun. Math. Phys.323(2013), 177–239

    2013

  14. [22]

    Gruber and J

    Ch. Gruber and J. Piasecki, Stationary motion of the adiabatic piston.Phys. A268(1999), 412–423

    1999

  15. [23]

    Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation.Comm

    Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation.Comm. Pure. Appl. Math.55(2006), no. 9, 0626–0687

    2006

  16. [24]

    Guo, Decay and continuity of Boltzmann equation in bounded domains.Arch

    Y. Guo, Decay and continuity of Boltzmann equation in bounded domains.Arch. Rat. Mech. Anal.197(2010), no. 3, 713–809

    2010

  17. [25]

    Y. Guo, C. Kim, D. Tonon and A. Trescases, Regularity of the Boltzmann equation in convex domains.Invent. Math.207 (2017), no. 1, 115–290. FREE BOUNDARY PROBLEM OF THE BOLTZMANN EQUATION 71

    2017

  18. [26]

    Kaltenbacher, I

    B. Kaltenbacher, I. Kukavica, I. Lasiecka, R. Triggiani, A. Tuffaha and J. Webster,Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions, Lecture notes from Oberwolfach seminars, November 20–26, 2016, Oberwolfach Semin. 48, Birkh¨ auser/Springer, 2018

    2016

  19. [27]

    Koike, Wall effect on the motion of a rigid body immersed in a free molecular flow.Kinet

    K. Koike, Wall effect on the motion of a rigid body immersed in a free molecular flow.Kinet. Relat. Models11(2018), 441–467

    2018

  20. [28]

    Koike, Motion of a rigid body in a special Lorentz gas: loss of memory effect.J

    K. Koike, Motion of a rigid body in a special Lorentz gas: loss of memory effect.J. Stat. Phys.172(2018), 795–823

    2018

  21. [29]

    Koike, Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions.J

    K. Koike, Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions.J. Differential Equations271(2021), 356–413

    2021

  22. [30]

    Koike, Time-asymptotic expansion with pointwise remainder estimates for 1D viscous compressible flow.Arch

    K. Koike, Time-asymptotic expansion with pointwise remainder estimates for 1D viscous compressible flow.Arch. Rational Mech. Anal.247(2023), 81

    2023

  23. [31]

    Khalil, Mesoscopic description of the adiabatic piston: kinetic equations and H-theorem.J

    N. Khalil, Mesoscopic description of the adiabatic piston: kinetic equations and H-theorem.J. Stat. Phys.176(2019), no. 5, 1138–1160

    2019

  24. [32]

    Li and Y

    F. Li and Y. Wang, Global strong solutions to the Vlasov-Poisson-Boltzmann system with soft potential in a bounded domain,J. Differential Equations.305(2021), 143–205

    2021

  25. [33]

    Sisti and C

    F. Sisti and C. Ricciuti, Effects of concavity on the motion of a body immersed in a Vlasov gas.SIAM J. Math. Anal.46 (2014), no. 6, 3579–3611

    2014

  26. [34]

    Sone,Molecular Gas Dynamics: Theory, Techniques, and Applications

    Y. Sone,Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkhauser, Boston, 2007

    2007

  27. [35]

    Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface.Arch

    A. Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface.Arch. Rational Mech. Anal.133(1996), no. 4, 299–331

    1996

  28. [36]

    Tsuji and K

    T. Tsuji and K. Aoki, Decay of a linear pendulum in a free-molecular gas and in a special Lorentz gas.J. Stat. Phys.146 (2012), no. 3, 620–645

    2012

  29. [37]

    Tsuji and K

    T. Tsuji and K. Aoki, Moving boundary problems for a rarefied gas: spatially one-dimensional case.J. Comput. Phys.250 (2013), 574–600

    2013

  30. [38]

    Tsuji and K

    T. Tsuji and K. Aoki, Decay of a linear pendulum in a collisional gas: Spatially one-dimensional case.Phys. Rev. E89 (2014), 052129

    2014

  31. [39]

    Tsuji, J

    T. Tsuji, J. Arai and S. Kawano, Slow approach to steady motion of a concave body in a free-molecular gas.Phys. Rev. E (3)92(2015), no. 1, 012130, 15 pp

    2015

  32. [40]

    Tsuji, S

    T. Tsuji, S. Saita, and S. Kawano, Thermophoresis of a Brownian particle driven by inhomogeneous thermal fluctuation. Phys. A493(2018), 467–482

    2018

  33. [41]

    R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadel- phia, PA, 1996

    1996

  34. [42]

    Ukai and T

    S. Ukai and T. Yang, The Boltzmann equation in the spaceL 2 ∩L ∞ β : Global and time-periodic solutions.Anal. Appl. (Singap.)4(2006), 263–310

    2006

  35. [43]

    J. L. V´ azquez and E. Zuazua, Large time behavior for a simplified 1D model of fluid-solid interaction.Commun. Partial Differ. Equ.28(2003), 1705–1738. (SCC)School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University...

    2003

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.