arXiv subjects

Pylyp Cherevan

Publications and source records attributed to Pylyp Cherevan.

4 recordsLinked to original sources

arXiv2025

An {\epsilon}-free rank-6 decoupling estimate for the paraboloid surface

For the paraboloid decomposition $F=\sum_{\Theta} F_{\Theta}$ with $\Theta\subset{|\xi|\sim\lambda}$ and radius $r=\lambda^{-2/3}$, we prove a log-free estimate $|F|{L^{6}(Q{\lambda})}\lesssim \lambda^{\Sigma_{\lambda}} D^{\Sigma_{D}} \big(\sum_{\Theta}|F_{\Theta}|{L^{6}}^{2}\big)^{1/2}$ as $\lambda\to\infty$, where $D=\lambda^{1/12}$. Key components: (i) broad geometry of rank 3: bilipschitz behavior of normals gives $\max{i c{} D$ brings a factor $D$ ($+1/12$ in $\lambda$, $+1$ in $D$); (iv) algebraic shell: excluding a neighborhood $N_{\beta}(P)$ contributes $-1/12$ in $\lambda$ and $-1$ in $D$; (v) tube packing: explanatory only; (vi) narrow cascade: a double $7/8$ rescaling exits the narrow regime and contributes $-5/64$ in $\lambda$ (zero in $D$). Summing exponents: $\Sigma_{\lambda}=5/36-9/2-5/64=-2557/576\approx -4.44<0$ and $\Sigma_{D}=-3+1-1=-3<0$, hence both $\lambda^{\varepsilon}$- and $D^{\varepsilon}$-losses are removed.

math.GM↗

arXiv2025

A log-free estimate for the diagonal paraproduct high $\times$ high $\to$ low in the 3D Navier-Stokes equation

We consider the diagonal paraproduct arising in the nonlinearity $(u\cdot \nabla) u$ for the three-dimensional Navier-Stokes equations. On scale-critical windows and in the range $1/6 < \delta \le 5/8$ we obtain a log-free estimate at the level $L^2_t {\dot H}^{-1}_x$ for the projection $P_{< N^{1-\delta}} \nabla(u_N \otimes v_N)$, consistent with the critical energy scheme. The main tools are phase-geometric integration, anisotropic local estimates on cylinders, and bilinear $ell^2$ decoupling on a finite-rank surface; the narrow diagonal zone is controlled via suppression of the null form. The work is restricted to a single resonant component; extensions to the full structure $(u \cdot\nabla) u$ and to sup$_t$ versions are left for further analysis.

math.AP↗

arXiv2025

Log-free estimate of the full nonlinearity in the three-dimensional Navier-Stokes equations outside the diagonal regime

We investigate the contribution of the full nonlinearity outside the narrow diagonal zone in the three-dimensional Navier-Stokes equations. We consider the off-diagonal components, including lh, hl, as well as part of the resonant block hh -> l for |xi + eta| >= N^(1-delta). The proof relies on three main elements: (i) six-fold integration by parts in the phase Phi(t,x,xi,eta) = x*(xi + eta) + 4trho1rho2 with respect to (t,rho1,rho2); on the window |t| <= N^(-1/2) the phase Hessian A = nabla^2_(t,rho1,rho2) Phi is non-degenerate and provides a reserve |det A| ~ N^(3/2 - delta); (ii) local Strichartz estimates on cylinders of scale N^(-1/2); in Sec. 4 a strengthened version is used to combine with the decoupling scheme, while the unconditional framework is based on heat reduction (App. D) and globalization (App. E); and (iii) bilinear epsilon-free decoupling in folded geometry of rank 4 (Appendix B), yielding a gain of N^(-1/4) for angular tiles of width N^(-1/2). For the narrow corona, suppression of the null-form type symbol is realized when delta > 1/2; for the block hh -> h with output projection P_N this mechanism is not required and is accounted for separately (see App. E.6). The combined count yields an a priori estimate without logarithmic losses in the norm L^1_t H^-1_x over the whole zone |xi + eta| >= N^(1 - delta) for delta in (1/3, 5/8]; the upper bound is imposed by the stability of the phase reserve |det A| ~ N^(3/2 - delta) >> 1 on the window |t| <= N^(-1/2). The full scheme and navigation through the sections are given in the text.

math.AP↗

arXiv2025

Log-free estimate for the resonant paraproduct in the 3D Navier-Stokes equations

We consider the resonant paraproduct (high-high $\to$ low regime) in the nonlinearity $(u\cdot\nabla)u$ for the three-dimensional Navier-Stokes equations. For sufficiently smooth, divergence-free u, we establish the a priori estimate without logarithmic loss $$\|R_N(u)\|{\dot H^{-1}} \lesssim N^{-1}\,\|u\|{\dot H^{1/2}}\,\|u\|_{\dot H^{1}},$$ with a constant independent of the dyadic frequency $N$. The proof combines phase-geometric integration by parts along an adapted frame, wave-packet discretization at scale $N^{-1/2}$, and an anisotropic Strichartz estimate on time windows of length $N^{-1/2}$. In the wide angular region we apply bilinear decoupling on a rank-3 phase surface; in the geometry at hand the minimal curvature yields a gain of order $N^{-1/6+o(1)}$ (with $o(1)\to 0$ as $N\to\infty$), which suffices to remove the logarithmic loss. The contribution from the narrow region is handled separately by an energy argument in $\dot H^{-1}$ using null-form suppression near the interaction diagonal. The resulting bound is scale-consistent and requires no smallness assumptions, only the divergence-free condition on $u$. The analysis is restricted to a single resonant component of the paraproduct; potential extensions are discussed.

math.AP↗