arXiv subjects

Gabriel S. Koch

Publications and source records attributed to Gabriel S. Koch.

10 recordsLinked to original sources

arXiv2023

Properties of Navier-Stokes mild solutions with initial data in subcritical Lorentz spaces

For initial data $f$ in a subcritical Lorentz space $L^{p,q}(\mathbb{R}^{n}) \hookrightarrow \dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$ ($n<p<\infty$, $1\leq q \leq \infty$), we prove results which imply in particular that a local in time mild Navier-Stokes solution cannot become unbounded in the $L^{p,q}(\mathbb{R}^{n})$-norm before it becomes unbounded in the norm of the larger subcritical Besov space $\dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$. In view of the known local theory in such large Besov spaces, this can be thought of as a `propagation of regularity' type of result; here, we provide a self-contained local theory (including scaling-appropriate `blow-up estimates', similar to those established by J. Leray in the Lebesgue setting) in the Lorentz setting, along with uniqueness results which imply such propagation of regularity. Our existence results are based on the method of T. Kato (1984) with Lorentz spaces replacing Lebesgue spaces throughout, and are given without any reference to the Besov framework. The uniqueness results are similarly self-contained, extending certain Lebesgue space methods of Fabes-Jones-Riviere (1972) to the Lorentz setting. We also establish certain continuity properties of the solutions which are constructed. (A more detailed abstract is provided in the article itself.)

math.AP↗

arXiv2023

Properties of Navier-Stokes mild solutions in sub-critical Besov spaces whose regularity exceeds the critical value by $\boldsymbol{ε\in(0,1)}$

We consider mild solutions to the Navier-Stokes initial-value problem which belong to certain ranges $Z_{p,q}^{s}(T,n):=\widetilde{L}^{1}(0,T;\dot{B}_{p,q}^{s+2}(\mathbb{R}^{n}))\cap\widetilde{L}^{\infty}(0,T;\dot{B}_{p,q}^{s}(\mathbb{R}^{n}))$ of Chemin-Lerner spaces. For $n=3$, $ε\in(0,1)$ and $f\in\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{3})$, Chemin and Gallagher (Tunis. J. Math., 2019) construct a local solution $u\in\cap_{T'\in(0,T_{f,ε}^{*})}Z_{\infty,\infty}^{-1+ε}(T',3)$ with maximal existence time ${T_{f,ε}^{*}\gtrsim_{φ,ε}{\|f\|}_{\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{3})}^{-2/ε}}$, where $φ$ is the cutoff function used to define the Littlewood-Paley projections. We improve on this result as follows: for $n\geq 1$, $ε\in(0,1)$, $s\in(-1,\infty)$, $p,q\in[1,\infty]$, and initial data $f\in\dot{B}_{p,q}^{s}(\mathbb{R}^{n})\cap\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{n})$, we prove that there exists a unique local solution $u\in\cap_{T'\in(0,T^*_f)}\left(Z_{p,q}^{s}(T',n)\cap Z_{\infty,\infty}^{-1+ε}(T',n)\right)$ which, along with its maximal existence time $T_{f}^{*}\in(0,\infty]$, is independent of $ε,s,p,q$. If $T_{f}^{*}$ is finite, then we have the blow-up estimate (with explicit dependence on $ε$) ${\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{n})}\gtrsim_φε(1-ε){(T_{f}^{*}-t)}^{-ε/2}$ for all $t\in(0,T_{f}^{*})$. The solution is unique among all solutions in the larger class $\cap_{T'\in(0,T_{f}^{*})}\cup_{α\in(2,\infty)}L^α(0,T';L^{\infty}(\mathbb{R}^{n}))$, and if $T_{f}^{*}<\infty$ then $u\notin L^{2}(0,T_{f}^{*};L^{\infty}(\mathbb{R}^{n}))$. We also establish additional properties of the solution, depending on the Besov spaces to which the initial data belongs.

math.AP↗

arXiv2022

Partial regularity for Navier-Stokes and liquid crystals inequalities without maximum principle

In 1985, V. Scheffer discussed partial regularity results for what he called solutions to the "Navier-Stokes inequality". These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. We extend this notion to a system considered by Fang-Hua Lin and Chun Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the "director field" $d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies an a priori maximum principle for $d$ which they use to establish partial regularity (specifically, $\mathcal{P}^{1}(\mathcal{S})=0$) of solutions. For the analogous "inequality" one loses this maximum principle, but here we nonetheless establish certain partial regularity results (namely $\mathcal{P}^{\frac 92 + δ}(\mathcal{S})=0$, so that in particular the putative singular set $\mathcal{S}$ has space-time Lebesgue measure zero). Under an additional assumption on $d$ for any fixed value of a certain parameter $σ\in (5,6)$ (which for $σ=6$ reduces precisely to the boundedness of $d$ used by Lin and Liu), we obtain the same partial regularity ($\mathcal{P}^{1}(\mathcal{S})=0$) as do Lin and Liu. In particular, we recover the partial regularity result ($\mathcal{P}^{1}(\mathcal{S})=0$) of Caffarelli-Kohn-Nirenberg (1982) for "suitable weak solutions" of the Navier-Stokes system, and we verify Scheffer's assertion that the same hold for solutions of the weaker "inequality" as well.

math.AP↗

arXiv2022

Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by $\boldsymbol{ε\in [1,2]}$

For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+ε}(\mathbb{R}^n)}\gtrsim_{φ,ε,(p\vee q\vee 2)}{(T-t)}^{-ε/2}$ for all $ε\in[1,2)$ and $p,q\in[1,\frac{n}{2-ε})$, where $s_{p}:=-1+\frac{n}{p}$ is the scaling-critical regularity, and $φ$ is the cutoff function used to define the Littlewood-Paley projections. For $ε=2$, we prove the same type of estimate but only for $q=1$: ${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^n)}\gtrsim_{φ,(p\vee 2)}{(T-t)}^{-1}$ for all $p\in [1,\infty)$. Under the additional restriction that $p,q\in[1,2]$ and $n=3$, these blowup estimates are implied by those first proved by Robinson, Sadowski and Silva (J. Math. Phys., 2012) for $p=q=2$ in the case $ε\in(1,2)$, and by McCormick, Olson, Robinson, Rodrigo, Vidal-López and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases $(ε,q)=(1,2)$ and $(ε,q)=(2,1)$.

math.AP↗

arXiv2022

Parabolic fractal dimension of forward-singularities for Navier-Stokes and liquid crystals inequalities

In 1985, V. Scheffer discussed partial regularity results for what he called solutions to the "Navier-Stokes inequality". These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. One may extend this notion to a system considered by F.-H. Lin and C. Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the 'director field' $d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies an a priori maximum principle for $d$, which is lost when one considers the analogous 'inequality'. In 2018, Q. Liu proved a partial regularity result for certain solutions to the Lin-Liu model in terms of the parabolic fractal dimension $\textrm{dim}_{\textrm{pf}}$, relying on the boundedness of $d$ coming from the maximum principle. Specifically, Q. Liu proves $\textrm{dim}_{\textrm{pf}}(Σ_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}$ for any compact $\mathcal{K}$, where $Σ_{-}$ is the set of 'forward-singular' space-time points, near which the solution blows up forwards in time. For solutions to the corresponding 'inequality', we prove here that, without any compensation for the lack of maximum principle, one has $\textrm{dim}_{\textrm{pf}}(Σ_{-} \cap \mathcal{K}) \leq \tfrac {55}{13}$. We also provide a range of criteria, including as just one example the boundedness of $d$, any one of which would furthermore imply that $\textrm{dim}_{\textrm{pf}}(Σ_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}$ for solutions to the inequality, just as Q. Liu proved for solutions to the Lin-Liu system.

math.AP↗

arXiv2022

Notions of solution and weak-strong uniqueness criteria for the Navier-Stokes equations in Lorentz spaces

For initial data $f\in L^{2}(\mathbb{R}^n)$ ($n\geq 2$), we prove that if $p\in(n,\infty]$, any solution $u\in L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{2}H_{x}^{1}\cap L_{t}^{\frac{2p}{p-n}}L_{x}^{p,\infty}$ to the Navier-Stokes equations satisfies the energy equality, and that such a solution $u$ is unique among all solutions $v\in L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{2}H_{x}^{1}$ satisfying the energy inequality. This extends well-known results due to G. Prodi (1959) and J. Serrin (1963), which treated the Lebesgue space $L_{x}^{p}$ rather than the larger Lorentz (and `weak Lebesgue') space $L_{x}^{p,\infty}$. In doing so, we also prove the equivalence of various notions of solutions in $L_{x}^{p,\infty}$, generalizing in particular a result proved for the Lebesgue setting in Fabes-Jones-Riviere (1972).

math.AP↗

arXiv2016

Blow-up of critical Besov norms at a potential Navier-Stokes singularity

We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space $\dot B^{-1+\frac 3p}_{p,q}(\mathbb{R}^3)$, with $3 0$, then the norm of the solution in that Besov space becomes unbounded at time $T$. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza, Seregin and Sverak (Uspekhi Mat. Nauk 58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in $L^3(\mathbb{R}^3)$. Our proof uses profile decompositions and is based on our previous work (Math. Ann. 355(4):1527--1559, 2013) which provided an alternative proof of the $L^3(\mathbb{R}^3)$ result. For very large values of $p$, an iterative method, which may be of independent interest, enables us to use some techniques from the $L^3(\mathbb{R}^3)$ setting. (To appear in Communications in Mathematical Physics.)

math.AP↗

arXiv2012

A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion

In this paper we continue to develop an alternative viewpoint on recent studies of Navier-Stokes regularity in critical spaces, a program which was started in the recent work by C. Kenig and the second author (Ann Inst H Poincaré Anal Non Linéaire 28(2):159-187, 2011). Specifically, we prove that strong solutions which remain bounded in the space $L^3(R^3)$ do not become singular in finite time, a known result established by Escauriaza, Seregin and Sverak (Uspekhi Mat Nauk 58(2(350)):3-44, 2003) in the context of suitable weak solutions. Here, we use the method of "critical elements" which was recently developed by Kenig and Merle to treat critical dispersive equations. Our main tool is a "profile decomposition" for the Navier-Stokes equations in critical Besov spaces which we develop here. As a byproduct of this tool, assuming a singularity-producing initial datum for Navier-Stokes exists in a critical Lebesgue or Besov space, we show there is one with minimal norm, generalizing a result of Rusin and Sverak (J Funct Anal 260(3):879-891, 2011).

math.AP↗

arXiv2010

An alternative approach to regularity for the Navier-Stokes equations in critical spaces

In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space $\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is the first instance in which this method has been applied to a parabolic equation. We remark that we have restricted our attention to a special case due only to a technical restriction, and plan to return to the general case (the $L^3$ setting) in a future publication.

math.AP↗

arXiv2010

Profile decompositions for critical Lebesgue and Besov space embeddings

Profile decompositions for "critical" Sobolev-type embeddings are established, allowing one to regain some compactness despite the non-compact nature of the embeddings. Such decompositions have wide applications to the regularity theory of nonlinear partial differential equations, and have typically been established for spaces with Hilbert structure. Following the method of S. Jaffard, we treat settings of spaces with only Banach structure by use of wavelet bases. This has particular applications to the regularity theory of the Navier-Stokes equations, where many natural settings are non-Hilbertian.

math.AP↗