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Dafermos' principle and Brenier's duality scheme for defocusing dispersive equations

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Pith's one-line read This paper establishes a Dafermos-style entropy principle for defocusing dispersive equations: while a strong solution exists, no subsolution can dissipate the total entropy earlier or faster.

desk verdict A genuinely novel abstract duality framework with a real but fixable gap: the advertised power-law PDE applications violate the paper's own C^2 hypothesis for exactly the generic parameter ranges. read the letter →

arxiv 2501.05389 v2 pith:HXOX7HOA submitted 2025-01-09 math.AP math-phmath.FAmath.MP

Dmitry Vorotnikov This is my paper · ORCID
classification math.APmath-phmath.FAmath.MP MSC 35D9935L9037K5847A5649Q99
keywords DafermosprincipleBrenierdualityschemedefocusingNLSnonlinearKlein-GordonequationgeneralizedKdVanisotropicOrliczspacesentropydissipationsubsolutions
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 aims to show that several defocusing dispersive equations — the nonlinear Schrödinger, nonlinear Klein–Gordon, and generalized KdV equations with power-law nonlinearities — fit a single abstract problem $\partial_t v = L(F(v))$ with a strictly convex, formally conserved entropy $K$. Inside that abstract problem the paper proves that a dual matrix-valued variational problem is consistent on large time intervals once the weight is allowed to decay exponentially. The main payoff is a no-early-dissipation principle: on any interval where a strong solution exists, no subsolution can dissipate the total entropy earlier or faster than the strong solution. If true, this gives an entropy-based selection statement for equations that generally admit many weak solutions, and the statement is new even for the quadratic flux $F(v)=v\otimes v$, hence for the incompressible Euler system.

What carries the argument

The central object is the dual matrix-valued variational problem built from the entropy $K(v)=\frac12\operatorname{Tr}(F(v)-F(0))$ and the sharp variable $v^\#=\nabla K(v)$. Because $K$ is an $N$-function satisfying the $\Delta_2$ condition together with its Legendre transform, the analysis lives in anisotropic Orlicz spaces $L^K$, and the sharp formulation of the equation is $\partial_t(v^\#)_l+L^*(v^\#):\partial_l F(\nabla K^*(v^\#))=0$. The dual problem maximizes $\int_0^T -(v_0,E)\,dt+\mathcal K(E,B)$, where $\mathcal K(E,B)$ is the infimum of $\int_0^T[(z,E)+\frac12(M,hI+2B)]\,dt$ over pairs with $F(z)\le M$, subject to $\partial_t B=L^*E$, $B(T)=0$, and $hI+2B\succcurlyeq 0$. The notion of $\Lambda$-convexity relaxes Loewner convexity by testing convexity only against matrices in the subspace $\Lambda$ generated by $L^*$, which is what lets non-quadratic dispersive examples fit. The time-adaptive weight $h(t)=e^{-\gamma t}$, with $H(t)=\int_T^t h(s)\,ds$, makes the condition $hI\succcurlyeq -2H L^*(v^\#)$ achievable on any $[0,T_1]$ with $T_1<T$, and that condition converts the strong solution into a dual maximizer without a duality gap. Uniqueness of strong solutions is carried by the Jeffreys divergence $J(t)=(u-v,u^\#-v^\#)$, whose time derivative is controlled by $\Lambda$-convexity and Grönwall's inequality.

What would settle it

Exhibit, on a torus, a smooth strong solution of the defocusing NLS (or GKdV) together with a subsolution $(u,M)$ in the sense of Definition 3.10 whose total entropy $\widetilde K(t)$ satisfies $\widetilde K(t)\le K(0)$ for almost all $t\in(0,t_1)$ and $\widetilde K(t)<K(0)$ for almost all $t\in(t_0,t_1)$; Theorem 4.3 explicitly forbids exactly this configuration, so its appearance would falsify the paper's central claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.3. Let $v$ be a strong solution of $\partial_t v = L(F(v))$ on $[0,T]$ with conserved total entropy $K(t)=K(0)$, and let $(u,M)$ be any subsolution with total entropy $\widetilde K(t)$. The theorem asserts that there are no $0\le t_0<t_1\le T$ for which $\widetilde K(t)\le K(t)$ for almost all $t\in(0,t_1)$ and $\widetilde K(t)<K(t)$ for almost all $t\in(t_0,t_1)$. In words, a subsolution cannot be strictly below the strong solution's entropy during an entire interval without having exceeded it earlier; any strict drop must be preceded by an excess. The theorem is derived from Theorem 4.1, which shows that for the time-adaptive weight $h(t)=e^{-\gamma t}$ the pair $(E_+,B_+)=(\partial_t(Hv^\#), L^*(Hv^\#))$ exactly maximizes the dual problem and closes the duality gap, with common value $H(0)K_0$. The strong solution is recovered from the dual maximizer by $v(t,x)=\nabla K^*\big((1/H(t))\int_t^T (-E_+)(s,x)\,ds\big)$. Remark 4.4 records that this principle is new even in the quadratic case $F(v)=v\otimes v$, and hence applies to the incompressible Euler system.

Load-bearing premise

The load-bearing premise is that a strong solution exists with the sharp-variable regularity of Definition 3.13: $\partial_t(Hv^\#)$ must lie in the dual Orlicz space and $H L^*(v^\#)$ must be essentially bounded; without that regularity, entropy conservation and the no-early-dissipation conclusion are not established.

Editorial extensions

If this is right

  • For the defocusing NLS, NLKG, and GKdV equations treated in Section 7, the no-early-dissipation principle holds on every interval inside the strong solution's existence time: a subsolution whose total entropy never exceeds the conserved value must equal it almost everywhere, so strict dissipation before the strong solution is impossible.
  • The dual variational problem has a maximizer in the relevant anisotropic Orlicz space for every admissible initial datum whenever the operator satisfies the strong trace condition, and the optimal value is finite; this yields global-in-time dual variational solutions for the dispersive examples without the usual restrictions on exponents or data size.
  • Strong solutions with the same initial datum are unique, by a Grönwall argument on the Jeffreys divergence.
  • Whenever a strong solution exists, solving the dual problem is equivalent to finding it: formula (4.4) reconstructs the strong solution from the dual maximizer.
  • The results cover the quadratic flux $F(v)=v\otimes v$, so the principle applies to the incompressible Euler system and the other quadratic examples from the author's earlier work.

Reading between the lines

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

  • Editorial extension: the time-adaptive weight mechanism suggests a general recipe for removing small-time restrictions in other dual variational schemes: any strong solution whose adjoint flux $H L^*(v^\#)$ is essentially bounded can be shielded by an exponentially decaying weight, so the consistency argument should transfer to non-quadratic systems beyond the three worked examples.
  • Editorial extension: the principle yields a numerical test. For a known smooth solution of defocusing NLS or GKdV, construct subsolutions with oscillatory corrections and compute their total entropy; the theorem predicts any strict dip below the conserved entropy must be preceded by a positive-measure interval where the subsolution's entropy exceeds the conserved value.
  • Editorial extension: the anisotropic Orlicz formulation points to entropy-modular bounds as the natural regularity measure for dispersive weak solutions, and the appendix's ballistic-transport analogy suggests the dual problem could be read as an optimal-transport problem on Orlicz–Wasserstein spaces; making that analogy rigorous is a plausible next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops an abstract duality framework, in the spirit of Brenier's matrix-valued variational formulations, for evolution equations of the form ∂_t v = L(F(v)). Under convexity, positivity, and formal conservativity assumptions on the matrix function F, it introduces an entropy K(v) = (1/2)Tr(F(v)-F(0)), defines weak solutions, subsolutions, and strong solutions in anisotropic Orlicz spaces, and proves several structural results: conservation of entropy for strong solutions (Lemma 3.16), consistency of the dual problem with no duality gap (Theorem 4.1), a Dafermos-type principle asserting that no subsolution can have total entropy strictly below that of a strong solution on a whole interval (Theorem 4.3), solvability of the dual problem in Orlicz spaces (Theorem 5.4), and uniqueness of strong solutions (Theorem 6.1). The final sections apply the abstract framework to scalar conservation laws, generalized KdV, defocusing NLS, and complex NLKG with power-law nonlinearities.

Significance. If fully established, the paper would be a substantial contribution: it provides a unified variational treatment of several nonlinear dispersive equations, extends Brenier's duality scheme beyond quadratic nonlinearities and small time intervals, and gives a conditional no-earlier-and-no-faster entropy dissipation principle that is new even in the quadratic/Euler setting. The abstract consistency theorem and the Orlicz-space solvability result are the main technical achievements and are presented with largely complete arguments, modulo the regularity caveats below. The paper is also honest about its conditional nature for the PDE applications, although the advertised ranges of power-law exponents are currently not justified by the stated hypotheses.

major comments (3)
  1. [§7.2–§7.3, §8; Section 3, Assumption 3.2] The applications do not satisfy the standing C^2-smoothness hypothesis for the ranges claimed. Section 3 fixes F as a C^2-smooth matrix function, but the NLS and NLKG fluxes in §7.3 and §8 contain entries proportional to (a^2+b^2)^q (through the term ε v̄⊗v̄), and the second derivatives of this function are unbounded near the origin whenever q<1. Similarly, the GKdV flux in §7.2 contains |u|^α, which is not C^2 (and for α=1 not even C^1) when α<2. The verification of Assumption 3.2 and of the derivative bound (3.6) in these sections is therefore invalid at the origin for the announced ranges q≥1/2 and α≥1. Since Lemma 3.16 and Theorems 4.1 and 4.3 rely on the abstract hypotheses, the claimed power-law applications are not established as stated. The statement would become correct if the ranges were restricted to q≥1 and α≥2, or if the abstract regularity assumptions were weakened and the applications reworked accordingly.
  2. [§7.2–§7.3, §8; Definition 3.13] The paper does not prove that the auxiliary extended systems admit strong solutions in the regularity class (3.18), (3.24) for the relevant initial data, nor does it derive this regularity from the original NLS/GKdV/NLKG Cauchy problems. Theorem 5.4 produces dual maximizers and Remark 5.5 defines only 'generalized solutions', but Theorem 4.3 concerns strong solutions. Consequently, the sentence in §7 that the theorems of Sections 4–6 are 'fully applicable here' overstates what is proved; the Dafermos principle for these equations remains conditional on the existence of a strong solution in the abstract sense.
  3. [§6, Theorem 6.1] The uniqueness theorem is proved only under an informal 'regular enough' assumption, and the general case is deferred with the phrase 'tedious and rather standard technicalities'. Given the genuinely low regularity of the class (3.18)–(3.24), including time-weights and anisotropic Orlicz spaces, the missing approximation argument is a nontrivial part of the proof. As written, Theorem 6.1 is a proof sketch rather than a complete proof.
minor comments (4)
  1. [§2, Propositions 2.9 and 2.11] Both propositions are stated with proofs omitted or delegated to the classical isotropic arguments. Since the anisotropic Orlicz setting is central to the paper, either the proofs should be included in the appendix or precise references for the anisotropic versions should be supplied.
  2. [§7.3, Λ-convexity verification] The verification of Λ-convexity for the NLS flux is summarized as 'tedious but elementary', and the explicit Hessian computation is not displayed. For the smooth range q≥1 this is acceptable, but the dependence of the admissible small constant ε on q should be made explicit, because it is part of the verification of Assumption 3.2.
  3. [§4, Eq. (4.13)] The bounds of integration in the displayed inequality (4.13) appear to be written in the wrong order; the intended expression should be an integral over (t0,T1) or the claim should be reformulated for clarity.
  4. [§4, Theorem 4.3] The wording 'no subsolution can dissipate the total entropy earlier or faster' is potentially confusing because the theorem excludes a subsolution whose total entropy is strictly below the strong solution's entropy on an interval. A more neutral phrasing, such as 'no subsolution can have total entropy strictly below that of the strong solution on a whole interval', would better match the statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Dafermos principle is proved conditionally from stated assumptions rather than being a restatement of its inputs.

full rationale

The paper's central result is a conditional theorem: assuming the existence of a strong solution v in the precise regularity class (3.18), (3.24), Lemma 3.16 derives conservation of the total entropy K(t), Theorem 4.1 establishes no duality gap for the dual pair (E+,B+) = (∂t(Hv#), H L*(v#)), and Theorem 4.3 derives the Dafermos statement by a contradiction argument against the subsolution infimum. None of these steps assumes the conclusion. The pair (E+,B+) is constructed from the strong solution, but its optimality is proved via Λ-convexity and a convex-Fenchel type inequality, not imposed by definition. No parameter is fitted to data and then renamed a prediction; the adaptive weight h(t)=exp(-γt) in Remark 4.2 is a proof device chosen to enforce (4.1) on any T1<T. The self-citations to [57] are contextual: the quadratic-case uniqueness implementation and the Euler-subsolution equivalence remark are not load-bearing for the main framework, which is proved in the text. The regularity gaps for the power-law applications (q<1 for NLS/NLKG and α<2 for GKdV) are hypothesis-check failures and therefore correctness risks, not circular reductions: the applications assume a strong solution of an auxiliary extended system and verify the structural assumptions formally. No exhibited step reduces an equation to itself by construction, and no fitted input is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The abstract theory rests on structural assumptions about F, L, K and the strong solution. No new physical particles, forces or fields are introduced. The sharp variable v# and the dual variational solutions are mathematical constructs, not independent empirical entities.

free parameters (2)
  • epsilon (auxiliary scaling in F and L) = small positive, chosen sufficiently small depending on α or q
    In Sections 7.2, 7.3 and 8, the flux F and operator L are constructed with a small parameter ε that cancels in the resulting PDE. It is chosen by hand to make F nonnegative and Λ-convex; no data are fitted to it.
  • gamma (adaptive weight exponent) = large positive, chosen in Remark 4.2 and Theorem 4.3
    The weight h(t)=exp(-γt) is selected after the strong solution is known, so that hI ⪰ -2H L*(v#) on the chosen interval. It is a proof device rather than an empirical parameter.
assumptions (7)
  • standard math Orlicz-space facts: N-functions, Δ2 condition, modular convergence, Nemytskii continuity, anisotropic Hardy inequality.
    These are invoked throughout Sections 2, 5 and Appendix A, with some proofs delegated to the cited literature.
  • domain assumption Assumption 3.1: K(v)=1/2 Tr(F(v)-F(0)) is a strictly convex N-function and K,K* satisfy the Δ2 condition.
    This defines the entropy and the Orlicz space for the whole theory.
  • domain assumption Assumptions 3.2 and 3.3: F is Λ-convex, F(R^n)⊂R_Λ, and L*(u(x)) belongs to Λ for test functions u.
    These are the convexity and positivity properties that make the duality scheme work; they are verified in examples but assumed in the abstract results.
  • domain assumption Assumption 3.7: formal conservativity (F(v), L*(v#))=0 for smooth v#.
    This conservation condition is essential for entropy conservation and for the consistency theorem. It is checked for GKdV, NLS and NLKG, but it is a nontrivial structural assumption.
  • domain assumption Existence of a strong solution in the regularity class (3.18),(3.24), plus the weight inequality (4.1) after choosing h.
    Theorems 4.1 and 4.3 are conditional on this existence and regularity. The paper does not prove such strong solutions for the examples.
  • domain assumption Strong trace condition for L in Definition 5.1, used for existence of dual maximizers.
    This is needed in Theorem 5.4 to convert the cone constraint into a uniform bound on B. It fails for the direct conservation-law formulation and is only satisfied in the examples after reformulation.
  • ad hoc to paper F is C²-smooth and the examples have smooth enough power nonlinearities.
    Assumption 3.2 says F is C², but the examples allow noninteger powers q≥1/2 and α≥1, for which functions such as (a²+b²)^q or |u|^α are not C² at the origin for q<1 or 1<α<2. This is an unflagged inconsistency in the applications.

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Cite this review

Pith. "Pith review of Dafermos' principle and Brenier's duality scheme for defocusing dispersive equations." pith.science (2026). https://pith.science/paper/HXOX7HOA

@misc{pith2026250105389,
  author       = {Pith},
  title        = {Pith review of: Dafermos' principle and Brenier's duality scheme for defocusing dispersive equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HXOX7HOA}},
  note         = {Machine review of arXiv:2501.05389}
}
read the original abstract

We discover an abstract structure behind several nonlinear dispersive equations (including the NLS, NLKG and GKdV equations with generic defocusing power-law nonlinearities) that is reminiscent of hyperbolic conservation laws. The underlying abstract problem admits an "entropy" that is formally conserved. The entropy is determined by a strictly convex function that naturally generates an anisotropic Orlicz space. For such problems, we introduce the dual matrix-valued variational formulation in the spirit of [Y. Brenier. Comm. Math. Phys. (2018) 364(2) 579-605]. Employing time-adaptive weights, we are able to prove consistency of the duality scheme on large time intervals. We also prove solvability of the dual problem in the corresponding anisotropic Orlicz spaces. As an application, we show that no subsolution of the PDEs that fit into our framework is able to dissipate the total entropy earlier or faster than the strong solution on the interval of existence of the latter. This result (we call it Dafermos' principle) is new even for "isotropic" problems such as the incompressible Euler system.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

Works this paper leans on

58 extracted references · 52 canonical work pages · cited by 1 Pith paper

  1. [1]

    A. Acharya. Variational principles for nonlinear PDE sy stems via duality. Quart. Appl. Math. , 81(1):127–140, 2023

    work page 2023

  2. [2]

    Acharya and A

    A. Acharya and A. N. Sengupta. Action principles for diss ipative, non-holonomic Newtonian mechanics. Proc. A., 480(2293):Paper No. 20240113, 21, 2024

    work page 2024

  3. [3]

    Acharya, B

    A. Acharya, B. Stroffolini, and A. Zarnescu. Variational dual solutions for incompressible fluids. arXiv preprint arXiv:2409.04911, 2024

    arXiv 2024

  4. [4]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient flows in metric spaces and in the space of probabilit y measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, second edition, 2008

    work page 2008

  5. [5]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savar´ e. Density of lipschi tz functions and equivalence of weak gradients in metric measure spaces. Revista Matem´ atica Iberoamericana, 29(3):969–996, 2013

    work page 2013

  6. [6]

    Baradat and L

    A. Baradat and L. Monsaingeon. Small noise limit and conv exity for generalized incompressible flows, Schr¨ odinger problems, and optimal transport. Arch. Ration. Mech. Anal. , 235(2):1357–1403, 2020

    work page 2020

  7. [7]

    Barton and N

    A. Barton and N. Ghoussoub. Dynamic and stochastic propa gation of the Brenier optimal mass trans- port. European Journal of Applied Mathematics , 30(6):1264–1299, 2019

    work page 2019

  8. [8]

    Benamou and Y

    J.-D. Benamou and Y. Brenier. A computational fluid mecha nics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. , 84(3):375–393, 2000

    work page 2000

Show all 58 references
  1. [9]

    J. M. Borwein and J. D. Vanderwerff. Convex functions: constructions, characterizations and c oun- terexamples, volume 109 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2010

    2010

  2. [10]

    Boulenger and E

    T. Boulenger and E. Lenzmann. Blowup for biharmonic NLS . Ann. Sci. ´Ec. Norm. Sup´ er. (4) , 50(3):503–544, 2017

    2017

  3. [11]

    Y. Brenier. The initial value problem for the Euler equa tions of incompressible fluids viewed as a concave maximization problem. Comm. Math. Phys. , 364(2):579–605, 2018

    2018

  4. [12]

    Y. Brenier. Examples of hidden convexity in nonlinear p des. Preprint, 2020

    2020

  5. [13]

    Brenier and I

    Y. Brenier and I. Moyano. Relaxed solutions for incompr essible inviscid flows: a variational and grav- itational approximation to the initial value problem. Philos. Trans. Roy. Soc. A , 380(2219):Paper No. 20210078, 12, 2022

    2022

  6. [14]

    A. C. Bronzi, M. C. Lopes Filho, and H. J. Nussenzveig Lop es. Wild solutions for 2d incompressible ideal flow with passive tracer. Communications in Mathematical Sciences , 13(5):1333–1343, 2015

    2015

  7. [15]

    Castro, D

    ´A. Castro, D. Faraco, and B. Gebhard. Entropy solutions to ma croscopic ipm. arXiv preprint arXiv:2309.03637, 2023

    2023

  8. [16]

    Chiodaroli and O

    E. Chiodaroli and O. Kreml. On the energy dissipation ra te of solutions to the compressible isentropic Euler system. Arch. Ration. Mech. Anal. , 214(3):1019–1049, 2014

    2014

  9. [17]

    Chlebicka, P

    I. Chlebicka, P. Gwiazda, A. ´Swierczewska-Gwiazda, and A. Wr´ oblewska-Kami´ nska.Partial differen- tial equations in anisotropic Musielak-Orlicz spaces . Springer Monographs in Mathematics. Springer, Cham, 2021

    2021

  10. [18]

    Cie´ slak and G

    T. Cie´ slak and G. Jamr´ oz. Maximal dissipation in Hunt er-Saxton equation for bounded energy initial data. Adv. Math. , 290:590–613, 2016

    2016

  11. [19]

    C. M. Dafermos. The entropy rate admissibility criteri on for solutions of hyperbolic conservation laws. J. Differential Equations , 14:202–212, 1973

    1973

  12. [20]

    C. M. Dafermos. Maximal dissipation in equations of evo lution. J. Differential Equations , 252(1):567– 587, 2012

    2012

  13. [21]

    Daneri and L

    S. Daneri and L. Sz´ ekelyhidi, Jr. Non-uniqueness and h -principle for H¨ older-continuous weak solutions of the Euler equations. Arch. Ration. Mech. Anal. , 224(2):471–514, 2017

    2017

  14. [22]

    De Lellis and L

    C. De Lellis and L. Sz´ ekelyhidi, Jr. On admissibility c riteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. , 195(1):225–260, 2010

    2010

  15. [23]

    de Lellis and L

    C. de Lellis and L. Sz´ ekelyhidi Jr. The Euler equations as a differential inclusion. Annals of mathematics, 170(3):1417–1436, 2009. DAFERMOS’ PRINCIPLE 39

    2009

  16. [24]

    R. J. DiPerna. Uniqueness of solutions to hyperbolic co nservation laws. Indiana Univ. Math. J. , 28(1):137–188, 1979

    1979

  17. [25]

    T. K. Donaldson and N. S. Trudinger. Orlicz-Sobolev spa ces and imbedding theorems. J. Functional Analysis, 8:52–75, 1971

    1971

  18. [26]

    Feireisl

    E. Feireisl. Maximal dissipation and well-posedness f or the compressible Euler system. J. Math. Fluid Mech., 16(3):447–461, 2014

    2014

  19. [27]

    Friedman, O

    I. Friedman, O. Ria˜ no, S. Roudenko, D. Son, and K. Yang. Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity. Nonlinearity, 36(1):584–635, 2023

    2023

  20. [28]

    Gebhard, J

    B. Gebhard, J. Hirsch, and J. J. Kolumb´ an. On a degenera te elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 41(6):1527– 1594, 2024

    2024

  21. [29]

    Gebhard and J

    B. Gebhard and J. J. Kolumb´ an. Relaxation of the Boussi nesq system and applications to the Rayleigh- Taylor instability. NoDEA Nonlinear Differential Equations Appl. , 29(1):Paper No. 7, 38, 2022

    2022

  22. [30]

    Gimperlein, M

    H. Gimperlein, M. Grinfeld, R. J. Knops, and M. Slemrod. The least action admissibility principle. arXiv preprint arXiv:2409.07191, 2024

    2024 arXiv

  23. [31]

    Glimm, D

    J. Glimm, D. Lazarev, and G.-Q. G. Chen. Maximum entropy production as a necessary admissibility condition for the fluid Navier–Stokes and Euler equations. SN Applied Sciences , 2:1–9, 2020

    2020

  24. [32]

    H¨ ofer and N

    F. H¨ ofer and N. A. Nikov. On growth of Sobolev norms for p eriodic nonlinear Schr¨ odinger and gener- alised Korteweg-de Vries equations under critical Gibbs dy namics. arXiv preprint arXiv:2412.08630, 2024

    2024

  25. [33]

    L. Hsiao. The entropy rate admissibility criterion in g as dynamics. J. Differential Equations , 38(2):226– 238, 1980

    1980

  26. [34]

    Jordan, D

    R. Jordan, D. Kinderlehrer, and F. Otto. The variationa l formulation of the Fokker-Planck equation. SIAM J. Math. Anal. , 29(1):1–17, 1998

    1998

  27. [35]

    S.-C. Klein. Using the Dafermos entropy rate criterion in numerical schemes. BIT, 62(4):1673–1701, 2022

    2022

  28. [36]

    S.-C. Klein. Stabilizing discontinuous Galerkin meth ods using Dafermos’ entropy rate criterion: I— One-dimensional conservation laws. J. Sci. Comput. , 95(2):Paper No. 55, 37, 2023

    2023

  29. [37]

    S.-C. Klein. Stabilizing discontinuous Galerkin meth ods using Dafermos’ entropy rate criterion: II— Systems of conservation laws and entropy inequality predic tors. J. Sci. Comput. , 100(2):Paper No. 42, 49, 2024

    2024

  30. [38]

    M. A. Krasnoselskii. Topological methods in the theory of nonlinear integral equ ations. The Macmillan Company, New York, 1964

    1964

  31. [39]

    M. A. Krasnoselskii and J. B. Rutickii. Convex functions and Orlicz spaces . P. Noordhoff Ltd., Gronin- gen, 1961

    1961

  32. [40]

    F. Kraus. ¨Uber konvexe Matrixfunktionen. Math. Z. , 41(1):18–42, 1936

    1936

  33. [41]

    Krejˇ c ´ ı and I

    P. Krejˇ c ´ ı and I. Straˇ skraba. A uniqueness criterion for the Riemann problem. Hiroshima Math. J. , 27(2):307–346, 1997

    1997

  34. [42]

    L´ eonard

    C. L´ eonard. From the Schr¨ odinger problem to the Monge -Kantorovich problem. J. Funct. Anal. , 262(4):1879–1920, 2012

    1920

  35. [43]

    Liero, A

    M. Liero, A. Mielke, and G. Savar´ e. Optimal entropy-tr ansport problems and a new Hellinger- Kantorovich distance between positive measures. Invent. Math. , 211(3):969–1117, 2018

    2018

  36. [44]

    Linares and G

    F. Linares and G. Ponce. Introduction to nonlinear dispersive equations . Springer, 2014

    2014

  37. [45]

    L¨ owner

    K. L¨ owner. ¨Uber monotone Matrixfunktionen. Math. Z. , 38(1):177–216, 1934

    1934

  38. [46]

    L. M. Martyushev and V. D. Seleznev. Maximum entropy pro duction principle in physics, chemistry and biology. Phys. Rep. , 426(1):1–45, 2006

    2006

  39. [47]

    Masaki and J.-i

    S. Masaki and J.-i. Segata. Existence of a minimal non-s cattering solution to the mass-subcritical generalized Korteweg–de Vries equation. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35(2):283–326, 2018. 40 D. VOROTNIKOV

    2018

  40. [48]

    Monsaingeon, L

    L. Monsaingeon, L. Tamanini, and D. Vorotnikov. The dyn amical Schr¨ odinger problem in abstract metric spaces. Adv. Math. , 426:Paper No. 109100, 2023

    2023

  41. [49]

    Muratori and G

    M. Muratori and G. Savar´ e. Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties. Journal of Functional Analysis , 278(4):108347, 2020

    2020

  42. [50]

    Prigogine

    I. Prigogine. Introduction to thermodynamics of irreversible processes . Interscience Publishers, New York-London, revised edition, 1961

    1961

  43. [51]

    Shenfeld

    Y. Shenfeld. Matrix displacement convexity along dens ity flows. Arch. Ration. Mech. Anal. , 248(5):Pa- per No. 74, 41, 2024

    2024

  44. [52]

    Singh, J

    S. Singh, J. Ginster, and A. Acharya. A hidden convexity of nonlinear elasticity. J. Elasticity , 156(3):975–1014, 2024

    2024

  45. [53]

    W. A. Strauss. Nonlinear scattering theory at low energ y. J. Functional Analysis , 41(1):110–133, 1981

    1981

  46. [54]

    K.-T. Sturm. Generalized orlicz spaces and wasserstei n distances for convex–concave scale functions. Bulletin des sciences math´ ematiques, 135(6-7):795–802, 2011

    2011

  47. [55]

    Sulem and P.-L

    C. Sulem and P.-L. Sulem. The nonlinear Schr¨ odinger equation: self-focusing and wa ve collapse . Springer Science & Business Media, 2007

    2007

  48. [56]

    C. Villani. Topics in optimal transportation . American Mathematical Soc., 2003

    2003

  49. [57]

    Vorotnikov

    D. Vorotnikov. Partial differential equations with qua dratic nonlinearities viewed as matrix-valued optimal ballistic transport problems. Arch. Ration. Mech. Anal. , 243(3):1653–1698, 2022

    2022

  50. [58]

    H. Ziegler. Some extremum principles in irreversible t hermodynamics with application to continuum mechanics. In Progress in solid mechanics, Vol. IV , pages 91–193. North-Holland, Amsterdam, 1963. (D. Vorotnikov) University of Coimbra, CMUC, Department of Mathematics, 3001 -5...

    1963

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.