Encephalomalacia/gliosis, deep venous thrombosis, and cancer in Arg393His antithrombin Hanoi and the potential impact of the β-amyloid precursor protein (APP) on thrombosis and cancer

Khue Vu Nguyen; Khue Vu Nguyen

doi:10.3934/Neuroscience.2022010

AIMS Neuroscience

2022, Volume 9, Issue 2: 175-215. doi: 10.3934/Neuroscience.2022010

Previous Article Next Article

Review

Encephalomalacia/gliosis, deep venous thrombosis, and cancer in Arg393His antithrombin Hanoi and the potential impact of the β-amyloid precursor protein (APP) on thrombosis and cancer

Khue Vu Nguyen ^{1,2
,
,}

Former Institution Attended:
1.
Department of Medicine, Biochemical Genetics and Metabolism, The Mitochondrial and Metabolic Disease Center, School of Medicine, University of California, San Diego, Building CTF, Room C-103, 214 Dickinson Street, San Diego, CA 92103-8467, USA
2.
Department of Pediatrics, University of California, San Diego, School of Medicine, San Diego, La Jolla, CA 92093-0830, USA

Received: 25 October 2021 Revised: 15 April 2022 Accepted: 18 April 2022 Published: 21 April 2022

A heterozygous Arg393His point mutation at the reactive site of antithrombin (AT) gene causing thrombosis in a Vietnamese patient is reported and named as Arg393His in AT-Hanoi. The present variant is characterized by a severe reduction of functionally active AT plasma concentration to 42% of normal resulting in multiple severe thrombotic events such as cerebral venous thrombosis (CVT) (encephalomalacia/gliosis), recurrent deep venous thrombosis (DVT) and the development of kidney cancer. Today the complexity of thrombophilia has grown with appreciation that multiple inherited and acquired risk factors may interact to result in a clinically thrombotic phenotype. This article focuses on the following issues: (1) pathophysiology and clinical conditions of Arg393His in AT-Hanoi; (2) “two way association” between cancer and thrombosis in which venous thromboembolism (VTE) can be both a presenting sign and a complication of cancer; (3) efficacy of anticoagulants used for the prevention of cancer-related thrombosis; (4) conditions of acquired risk factors such as cancer or genetic disorders via epigenetic modifications in gene-gene (epistasis) and/or gene-environment interactions such as in Lesch-Nyhan disease (LND), in which the β-amyloid precursor protein (APP) that may interact to predispose a patient to thrombosis and cancer. It is also necessary to study the hypoxanthine-guanine phosphoribosyltransferase (HGprt) enzyme, AT, and APP using expression vectors for exploring their impact on LND, thrombosis as well as other human diseases, especially the ones related to APP such as Alzheimer's disease (AD) and cancer. For such a purpose, the construction of expression vectors for HGprt and APP, with or without the glycosyl-phosphatidylinositol (GPI) anchor, was performed as described in Ref. #148 (Nguyen, K. V., Naviaux, R. K., Nyhan, W. L. Lesch-Nyhan disease: I. Construction of expression vectors for hypoxanthine-guanine phosphoribosyltransferase (HGprt) enzyme and amyloid precursor protein (APP). Nucleosides Nucleotides Nucleic Acids 2020, 39: 905–922). In the same manner, the construction of expression vectors for AT and APP can be performed as shown in Figure 6. These expressions vectors, with or without GPI anchor, could be used as tools for (a) studying the effects of Arg393His mutation in AT; (b) studying the emerging role of Arg393His mutation in AT and cancer; (c) studying intermolecular interactions between APP and AT.

Furthermore, the construction of expression vectors as described in Ref. #148, especially the one with GPI, can be used as a model for the construction of expression vectors for any protein targeting to the cell plasma membrane for studying intermolecular interactions and could be therefore useful in the vaccines as well as antiviral drugs development (studying intermolecular interactions between the spike glycoprotein of the severe acute respiratory syndrome coronavirus 2, SARS-CoV-2, as well as its variants and the angiotensin-converting enzyme 2, ACE2, in coronavirus disease 2019 (COVID-19) [155],[156], for example).

Keywords:

Citation: Khue Vu Nguyen. Encephalomalacia/gliosis, deep venous thrombosis, and cancer in Arg393His antithrombin Hanoi and the potential impact of the β-amyloid precursor protein (APP) on thrombosis and cancer[J]. AIMS Neuroscience, 2022, 9(2): 175-215. doi: 10.3934/Neuroscience.2022010

Related Papers:

[1]	Khaled Kefi, Nasser S. Albalawi . Three weak solutions for degenerate weighted quasilinear elliptic equations with indefinite weights and variable exponents. AIMS Mathematics, 2025, 10(2): 4492-4503. doi: 10.3934/math.2025207
[2]	Shulin Zhang . Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations. AIMS Mathematics, 2022, 7(1): 1015-1034. doi: 10.3934/math.2022061
[3]	Ye Xue, Yongzhen Ge, Yunlan Wei . The existence of a nontrivial solution to an elliptic equation with critical Sobolev exponent and a general potential well. AIMS Mathematics, 2025, 10(3): 7339-7354. doi: 10.3934/math.2025336
[4]	Lixiong Wang, Ting Liu . Existence and regularity results for critical $(p, 2)$ -Laplacian equation. AIMS Mathematics, 2024, 9(11): 30186-30213. doi: 10.3934/math.20241458
[5]	Rui Sun . Soliton solutions for a class of generalized quasilinear Schrödinger equations. AIMS Mathematics, 2021, 6(9): 9660-9674. doi: 10.3934/math.2021563
[6]	Liang Xue, Jiafa Xu, Donal O'Regan . Positive solutions for a critical quasilinear Schrödinger equation. AIMS Mathematics, 2023, 8(8): 19566-19581. doi: 10.3934/math.2023998
[7]	Wangjin Yao . Variational approach to non-instantaneous impulsive differential equations with $p$ -Laplacian operator. AIMS Mathematics, 2022, 7(9): 17269-17285. doi: 10.3934/math.2022951
[8]	Xiaojie Guo, Zhiqing Han . Existence of solutions to a generalized quasilinear Schrödinger equation with concave-convex nonlinearities and potentials vanishing at infinity. AIMS Mathematics, 2023, 8(11): 27684-27711. doi: 10.3934/math.20231417
[9]	Nicky K. Tumalun, Philotheus E. A. Tuerah, Marvel G. Maukar, Anetha L. F. Tilaar, Patricia V. J. Runtu . An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations. AIMS Mathematics, 2023, 8(11): 26007-26020. doi: 10.3934/math.20231325
[10]	Xicuo Zha, Shuibo Huang, Qiaoyu Tian . Uniform boundedness results of solutions to mixed local and nonlocal elliptic operator. AIMS Mathematics, 2023, 8(9): 20665-20678. doi: 10.3934/math.20231053

Abstract

Abbreviations

Aβ:

β-amyloid peptide;

ACE2:

angiotensin-converting enzyme 2;

ACT:

activated clotting time;

AD:

Alzheimer's disease;

APC:

activated protein C;

aPL:

antiphospholipid;

APLP1:

β-amyloid precursor-like protein 1;

APLP2:

β-amyloid precursor-like protein 2;

APP:

β-amyloid precursor protein;

APP-mRNA:

β-amyloid precursor protein messenger RNA;

APPsα:

soluble APP fragment released from APP following the cleavage by α-secretase;

aPTT or APTT:

activated partial thromboplastin time;

AS:

alternative splicing;

ASCO:

American society of clinical oncology;

AT:

anthithrombin;

AT I:

antithrombin I;

AT II:

antithrombin II;

AT III:

antithrombin III;

AT IV:

antithrombin IV;

ATP:

adenosine-5′-triphosphate;

CAPS:

catastrophic antiphospholipid syndrome;

cdk2:

cyclin-dependent kinase 2;

COVID-19:

coronavirus disease 2019;

CNS:

central nervous system;

CT:

computed tomography;

CTA:

computed tomography angiography;

CUS:

compression ultrasound;

CVT:

cerebral venous thrombosis;

DVT:

deep venous thrombosis;

EGFR-R776H:

epidermal growth factor receptor-Arg776His;

FFP:

fresh frozen plasma;

GPI:

glycosyl-phosphatidylinositol;

GTP:

guanosine-5′-triphosphate;

HBS:

heparin binding site;

HDM2:

human homologue of the murine double minute 2 protein;

HGprt:

hypoxanthine-guanine phosphoribosyltransferase;

HIT:

heparin-induced thrombocytopenia;

HIV:

human immunodeficiency virus;

HND:

HGprt-related neurological dysfunction;

HPRT1:

hypoxanthine phosphoribosyltransferase 1;

HPRT1-mRNA:

hypoxanthine phosphoryltransferase 1 messenger RNA;

HRH:

HGprt-related hyperuricemia;

INDELS:

deletion followed by an insertion;

INR:

international normalized ratio;

KPI:

Kunitz protease inhibitor;

LMWH:

low-molecular-weight heparin;

LND:

Lesch-Nyhan disease;

LNV:

Lesch-Nyhan variant;

MIM:

Mendelian inheritance in man;

mRNA:

messenger RNA;

NET:

neutrophil extracellular trap;

NOAC:

non-vitamin K antagonist oral anticoagulant;

NSAID:

nonsteroidal anti-inflammatory drug;

P21/WAF1:

also known as cyclin-dependent kinase inhibitor 1 or CDK-interacting; protein 1, is a cyclin-dependent kinase inhibitor (CKI) that is capable; of inhibiting all cyclin/CDK complexes, and thus function as a regulator; of cell cycle progression at G1 and S phase;

PCR:

polymerase chain reaction;

PD:

pharmacodynamics;

PE:

pulmonary embolism;

pHi:

pH intracellular;

PK:

pharmacokinetics;

PN-2:

protease nexin-2;

PT:

prothrombin time;

PTT:

partial thromboplastin time;

RS:

reactive site;

SARS-CoV-2:

severe acute respiratory syndrome coronavirus 2;

siRNA:

small interfering RNA;

TAD:

trans-activation domain;

TF:

tissue factor;

TFPI:

tissue factor pathway inhibitor;

TK1:

thymidine kinase 1;

TP53:

tumor suppressor protein 53;

UFH:

unfractionated heparin;

VKA:

vitamin K antagonist;

VTE:

venous thromboembolism;

WARF:

Wisconsin alumni research foundation;

WNV:

West Nile virus

1. Introduction

Nematic liquid crystals are aggregates of molecules which possess same orientational order and are made of elongated, rod-like molecules. Hence, in the study of nematic liquid crystals, one approach is to consider the behavior of the director field $d$ in the absence of the velocity fields. Unfortunately, the flow velocity does disturb the alignment of the molecules. More importantly, the converse is also true, that is, a change in the alignment will induce velocity. This velocity will in turn affect the time evolution of the director field. In this process, we cannot assume that the velocity field will remain small even when we start with zero velocity field.

In the 1960's, Ericksen ^[3,4] and Leslie ^[10,11] developed the hydrodynamic theory of liquid crystals. The Ericksen-Leslie system consists of the following equations ^[12]:

$\begin{equation} \left\{ \begin{aligned} \rho_t+\nabla\cdot(\rho u) = &0, \\ (\rho u)_t = &\rho F_i+\sigma_{ji,j},\\ \rho_1(\omega_i)_t = &\rho_1G_i+g_i+\pi_{ji,j} , \end{aligned} \right. \end{equation}$

(1.1)

where (1.1) $_1$ , (1.1) $_2$ and (1.1) $_3$ , represent the conservation of mass, linear momentum and angular momentum respectively. Besides, $\rho$ denotes the fluid density, $u = (u_1, u_2, u_3)$ is the velocity vector and $d = (d_1, d_2, d_3)$ the direction vector,

$\begin{equation} \left\{ \begin{aligned} &\sigma_{ji} = -P\delta_{ij}-\rho\frac{\partial F}{\partial d_{k,j}}+\hat{\sigma}'_{ji},\\ &\pi_{ji} = \beta_jd_i+\rho\frac{\partial F}{\partial d_{i,j}},\\ &g_i = \gamma d_i-\beta_jd_{i,j}-\rho\frac{\partial F}{\partial d_i}+\hat{g}'_i, \end{aligned} \right. \end{equation}$

(1.2)

where $F_i$ is the external body force, $G_i$ denotes the external director body force and $\beta$ , $\gamma$ come from the restriction of the direction vector $|d| = 1$ . The following relations also hold:

$\begin{equation} \left\{ \begin{aligned} &2\rho F = k_{22}d_{i,j}d_{i,j}+(k_{11}-k_{22}-k_{24})d_{i,i}d_{j,j}+(k_{33}-k_{22})d_id_jd_{k,i}d_{kj}+k_{24}d_{i,j}d_{j,i},\\ &\hat{\sigma}'_{ji} = \mu_1d_kd_pA_{kp}d_id_j+\mu_2d_jN_i+\mu_3d_iN_j+\mu_4A_{ij}+\mu_5d_jd_kA_{ki}+\mu_6d_id_kA_{kj}, \\&\hat{g}_i = \lambda_1N_i+\lambda_2d_jA_{ji}, \end{aligned} \right. \end{equation}$

(1.3)

and

$\begin{equation} \left\{ \begin{aligned} \omega_i = &\dot{d}_i = \frac{\partial d_i}{\partial t}+u\cdot\nabla d_i,\\ N_i = &\omega_i+\omega_{ki}d_k,\\ N_{ij} = &\omega_{i,j}+\omega_{ki}d_{k,j}, \end{aligned} \right. \end{equation}$

(1.4)

where

$2A_{ij} = u_{i,j}+u_{j,i},\quad 2\omega_{i,j} = u_{i,j}-u_{j,i}.$

On the basis of the second law of thermodynamics and Onsager reciprocal relation, one obtain

$\lambda_1 = \mu_2-\mu_3,\quad\lambda_2 = \mu_5-\mu_6 = -(\mu_2+\mu_3).$

The nonlinear constraint $|d| = 1$ can also be relaxed by using the Ginzburg-Landau approximation, that is, instead of the restriction $|d| = 1$ , we add the term $\frac1{\varepsilon^2}(|d|^2-1)^2$ in $\rho F$ . In addition, to further simplify the calculation, one take $\rho_1 = 0$ , $\beta_j = 0$ , $\gamma = 0$ , $F_i = 0$ and $\rho F = |\nabla d|^2+\frac1{\varepsilon^2}(|d|^2-1)^2$ , choose the domain $\Omega = \mathbb{R}^3$ , obtain the simplified model of nematic liquid crystals:

$\begin{equation} \left\{ \begin{aligned} &\rho_t+\nabla\cdot(\rho u) = 0, \\ &(\rho u)_t+\nabla\cdot(\rho u\otimes u)-\mu\Delta u-(\mu+\lambda)\nabla \nabla\cdot u+\nabla p(\rho) = -\Delta d\cdot\nabla d,\\ &d_t+u\cdot\nabla d = \Delta d-f(d), \end{aligned} \right. \end{equation}$

(1.5)

with the following initial conditions

$\begin{equation} \rho(x,0) = \rho_0(x),\quad u(x,0) = u_0(x),\quad d(x,0) = d_0(x),\quad|d_0(x)| = 1, \end{equation}$

(1.6)

and

$\begin{equation} \rho_0-\bar{\rho}\in H^N(\mathbb{R}^3),\quad u_0\in H^N(\mathbb{R}^3),\quad d_0-\omega_0 \in H^N(\mathbb{R}^3), \end{equation}$

(1.7)

for any integer $N\geq 3$ with a fixed vector $\omega_0\in \mathbb{S}^2$ , that is, $|\omega_0| = 1$ . In this paper, we assume that $f(d) = \frac1{\varepsilon^2}(|d|^2-1)d$ ( $\varepsilon > 0$ ) is the Ginzburg-Landau approximation and the pressure $p = p(\rho)$ is a smooth function in a neighborhood of $\bar{\rho}$ with $p'(\bar{\rho}) > 0$ for $\bar{\rho} > 0$ . Moreover, $\mu$ and $\lambda$ are the shear viscosity and the bulk viscosity coefficients of the fluid, respectively. As usual, the following inequalities hold:

$\mu > 0,\quad 3\lambda+2\mu\geq0.$

The study of liquid crystals can be traced back to Ericksen ^[3,4] and Leslie ^[10,11] in the 1960s. Since then, there is a huge amount of literature on this topic. For the incompressible case, we refer the author to ^{[2,6,12,13,20]} and the reference therein. There are also many papers related to the compressible case, see for instance, ^{[1,5,7,8,17,19]} and the reference cited therein.

In ^[19], the authors rewrote system (1.5) in the perturbation form as

$\begin{equation} \left\{ \begin{aligned} &\varrho_t+\bar{\rho}\nabla\cdot u = -\varrho\nabla\cdot u-u\cdot\nabla\varrho, \\ &u_t-\bar{\mu}\Delta u-(\bar{\mu}+\bar{\lambda})\nabla\nabla\cdot u+\gamma\bar{\rho}\nabla\varrho \\&\quad\quad\quad\quad = -u\cdot\nabla u-h(\varrho)(\bar{\mu}\Delta u+(\bar{\mu}+\bar{\lambda})\nabla\nabla\cdot u)-g(\varrho)\nabla\varrho-\phi(\Delta d\cdot\nabla d),\\ &d_t+u\cdot\nabla d = \Delta d-f(d), \end{aligned} \right. \end{equation}$

(1.8)

where $\varrho = \rho-\bar{\rho}$ , $\bar{\mu} = \frac{\mu}{\bar{\rho}}$ , $\bar{\lambda} = \frac{\lambda}{\bar{\rho}}$ , $\gamma = \frac{p'(\bar{\rho})}{\bar{\rho^2}}$ and the nonlinear functions of $\varrho$ are defined by

$h(\varrho) = \frac{\varrho}{\varrho+\bar{\rho}},\quad g(\varrho) = \frac{p'(\varrho+\bar{\rho})}{\varrho+\bar{\rho}}-\frac{p'(\bar{\rho})}{\bar{\rho}},\quad\phi(\varrho) = \frac1{\rho+\bar{\rho}}.$

We remark that the functions $h(\varrho)$ , $g(\varrho)$ and $\phi(\varrho)$ satisfy (see ^[19])

$\begin{equation} |h(\varrho)|,|g(\varrho)|\leq C|\varrho|,\quad|\phi^{(l)}(\varrho)|,|h^{(k)}(\varrho)|,|g^{(k)}(\varrho)|\leq C\; \hbox{for any}\; l\geq0,\; k\geq1. \end{equation}$

(1.9)

Wei, Li and Yao ^[19] obtained the small initial data global well-posedness provided that $\|\varrho_0\|_{H^3}+\|u_0\|_{H^3}+\|d_0-\omega_0\|_{H^4}$ is sufficiently small. Moreover, the authors also showed the optimal decay rates of higher order spatial derivatives of of strong solutions provided that $(\varrho_0, u_0, \nabla d_0)\in \dot{H}^{-s}$ for some $s\in[0, \frac12]$ .

Next, we introduce the main results in ^[19]:

Lemma 1.1. (Small initial data global well-posedness ^[19]) Assume that $N\geq3$ and $(\varrho_0, u_0, d_0-\omega_0)\in H^N(\mathbb{R}^3)\times H^N(\mathbb{R}^3)\times H^{N+1}(\mathbb{R}^3)$ . Then for a unit vector $\omega_0$ , there exists a positive constant $\delta_0$ such that if

$\begin{equation} \|\varrho_0\|_{H^3}+\|u_0\|_{H^3}+\|d_0-\omega_0\|_{H^4}\leq\delta_0, \end{equation}$

(1.10)

then problem (1.8) has a unique global solution $(\varrho(t), u(t), d(t))$ satisfying that for all $t\geq0$ ,

$\begin{equation} \begin{aligned}& \frac d{dt}\left(\|\varrho\|^2_{H^N}+\|u\|_{H^N}^2+\|\nabla d\|_{H^N}^2\right) + C_0(\|\nabla\varrho\|^2_{H^{N-1}}+\|\nabla u\|_{H^N}^2+\|\nabla\nabla d\|_{H^N}^2)\leq0.\end{aligned} \end{equation}$

(1.11)

Lemma 1.2. (Decay estimates ^[19]) Assume that all the assumptions of Lemma 1.1 hold. Then, if $(\varrho_0, u_0, \nabla d_0)\in \dot{H}^{-s}$ for some $s\in[0, \frac12]$ , we have

$\begin{equation} \|\Lambda^{-s}\varrho(t)\|^2_{L^2}+\|\Lambda^{-s}u(t)\|_{L^2}^2+\|\Lambda^{-s}\nabla d(t)\|_{L^2}^2\leq C_1,\quad\forall t\geq0, \end{equation}$

(1.12)

and

$\begin{equation} \|\nabla^l\varrho\|_{H^{N-l}}+\|\nabla^lu\|_{H^{N-l}}+\|\nabla^{l+1}d\|_{H^{N-l}}\leq C_2(1+t)^{-\frac{l+s}2},\quad\forall t\geq0\; {and}\; l = 0,1,\cdots,N-1. \end{equation}$

(1.13)

The main purpose of this paper is to improve the decay results in ^[19]. First, we give a remark on the symbol stipulations of this paper.

Remark 1.3. In this paper, we use $H^k(\mathbb{R}^3)$ $(k\in\mathbb{R})$ , to denote the usual Sobolev spaces with norm $\|\cdot\|_{H^s}$ , and $L^p(\mathbb{R}^3)$ $(1\leq p\leq\infty)$ to denote the usual $L^p$ spaces with norm $\|\cdot\|_{L^p}$ . We also introduce the homogeneous negative index Sobolev space $\dot{H}^{-s}(\mathbb{R}^3)$ :

$\dot{H}^{-s}(\mathbb{R}^3): = \{f\in L^2(\mathbb{R}^3):\||\xi|^{-s}\hat{f}(\xi)\|_{L^2} < \infty\}$

endowed with the norm $\|f\|_{\dot{H}^{-s}}: = \||\xi|^{-s}\hat{f}(\xi)\|_{L^2}$ . The symbol $\nabla^l$ with an integer $l\geq0$ stands for the usual spatial derivatives of order $l$ . For instance, we define

$\nabla^lz = \{\partial_x^{\alpha}z_i||\alpha| = l,i = 1,2,3\},\; \; \; z = (z_1,z_2,z_3).$

If $l < 0$ or $l$ is not a positive integer, $\nabla^l$ stands for $\Lambda^l$ defined by

$\begin{equation} \nonumber \Lambda^sf(x) = \int_{\mathbb{R}^3}|\xi|^s\hat{f}(\xi)e^{2\pi ix\cdot\xi}d\xi, \end{equation}$

where $\hat{f}$ is the Fourier transform of $f$ . Besides, $C$ and $C_i$ ( $i = 0, 1, 2, \cdots$ ) will represent generic positive constants that may change from line to line even if in the same inequality. The notation $A\lesssim B$ means that $A\leq CB$ for a universal constant $C > 0$ that only depends on the parameters coming from the problem.

It is worth pointing out that in ^[19], the authors consider problem (1.5) in 3D case, the negative Sobolev norms were shown to be preserved along time evolution and enhance the decay rates. However, because the Ginzburg-Landau approximation term is difficulty to control, only $s\in[0, \frac12]$ were considered in ^[19]. In this paper, we ovcome the difficult caused by Ginzburg-Landau approximation, assume that $s\in[0, \frac32)$ , obtain the optimal decay rates of higher order spatial derivatives of strong solutions for problem (1.5). Our main results are stated in the following theorem.

Theorem 1.4. Assume that all the assumptions of Lemma 1.1 hold. Then, if $(\varrho_0, u_0, \nabla d_0)\in \dot{H}^{-s}$ for some $s\in[0, \frac32)$ , we have

$\begin{equation} \|\Lambda^{-s}\varrho(t)\|^2_{L^2}+\|\Lambda^{-s}u(t)\|_{L^2}^2+\|\Lambda^{-s}\nabla d(t)\|_{L^2}^2\leq C_0,\quad\forall t\geq0, \end{equation}$

(1.14)

and

$\begin{equation} \|\nabla^l\varrho\|_{H^{N-l}}+\|\nabla^lu\|_{H^{N-l}}+\|\nabla^{l+1}d\|_{H^{N-l}}\leq C (1+t)^{-\frac{l+s}2},\quad{for}\; l = 0, 1,\cdots,N-1,\; \; \forall t\geq0. \end{equation}$

(1.15)

Note that the Hardy-Littlewood-Sobolev theorem implies that for $p\in(1, 2]$ , $L^{p}(\mathbb{R}^3)\subset\dot{H}^{-s}(\mathbb{R}^3)$ with $s = 3(\frac1p-\frac12)\in[0, \frac32)$ . Then, on the basis of Lemma 1.2 and Theorem 1.4, we obtain the optimal decay estimates for system (1.8).

Corollary 1.5. Under the assumptions of Lemma 1.2 and Theorem 1.4, if we replace the $\dot{H}^{-s}(\mathbb{R}^3)$ assumption by

$(\varrho,u_0, \nabla{d}_0)\in L^{p}(\mathbb{R}^3),\quad 1 < p\leq 2,$

then for $l = 0, 1, \cdots, N-1$ , , the following decay estimate holds:

$\begin{equation} \begin{aligned} \|\nabla^l\varrho\|_{H^{N-l}}+\|\nabla^lu\|_{H^{N-l}}+\|\nabla^{l+1}d\|_{H^{N-l}}\leq C (1+t)^{-\left[\frac32\left(\frac1p -\frac12\right)+\frac l2 \right]},\quad\forall t\geq0 .\end{aligned} \end{equation}$

(1.16)

Remark 1.6. Lemma 1.1 shows the global well-posedness of strong solutions for system (1.5) provided that the smallness assumption (1.10) holds. One can use the energy method to obtain the higher order energy estimates for the solution to prove this lemma (see ^[19]). We remark that the negative Sobolev norm estimates did not appear in the proving process of Lemma 1.1, it is only used in the decay estimates. Hence, the value of $s$ in Lemma 1.2 and Theorem 1.4 do not affect the energy estimates (1.10) and (1.11). And those two estimates hold for both Lemma 1.2 and Theorem 1.4.

Remark 1.7. The main purpose of this paper is to prove Theorem 1.4 and Corollary 1.5 on the asymptotic behavior of strong solutions for a compressible Ericksen-Leslie system. We remark that the global well-posedness and asymptotic behavior of solutions are important for the study of nematic liquid crystals system. Thanks to the above properties of solutions, one can understand the model more profoundly. Our results maybe useful for the study of nematic liquid crystals.

The structure of this paper is organized as follows. In Section 2, we introduce some preliminary results. The proof of Theorem 1.4 is postponed in Section 3.

2. Preliminaries

We first show a useful Sobolev embedding theorem in the following Lemma 2.1:

Lemma 2.1. (^[15]) If $0\leq s < \frac32$ , one have

$\begin{equation} \|u\|_{L^{\frac6{3-2s}}(\mathbb{R}^3)}\lesssim\|u\|_{\dot{H}^s(\mathbb{R}^3)}\quad{for\; all}\; \; u\in \dot{H}^s(\mathbb{R}^3). \end{equation}$

(2.1)

In ^[14], the author proved the following Gagliardo-Nirenberg inequality:

Lemma 2.2. (^[14]) Let $0\leq m, \alpha\leq l$ , then we have

$\begin{equation} \|\nabla^{\alpha}f\|_{L^p(\mathbb{R}^3)}\lesssim\|\nabla^mf\|_{L^q(\mathbb{R}^3)}^{1-\theta}\|\nabla^lf\|_{L^r(\mathbb{R}^3)}^{\theta}, \end{equation}$

(2.2)

where $\theta\in[0, 1]$ and $\alpha$ satisfies

$\begin{equation} \frac{\alpha}3-\frac1p = \left(\frac m3-\frac1q\right)(1-\theta)+\left(\frac l3-\frac1r\right)\theta. \end{equation}$

(2.3)

Here, when $p = \infty$ , we require that $0 < \theta < 1$ .

One also introduce the Kato-Ponce inequality which is of great importance in our paper.

Lemma 2.3. (^[9]) Let $1 < p < \infty$ , $s > 0$ . There exists a positive constant $C$ such that

$\begin{equation} \|\nabla^s(fg)\|_{L^p(\mathbb{R}^3)}\lesssim \|f\|_{L^{p_1}(\mathbb{R}^3)}\|\nabla^sg\|_{L^{p_2}(\mathbb{R}^3)}+\|\nabla^sf\|_{L^{q_1}(\mathbb{R}^3)}\|g\|_{L^{q_2}(\mathbb{R}^3)}, \end{equation}$

(2.4)

where $p_2, q_2\in(1, \infty)$ satisfying $\frac1p = \frac1{p_1}+\frac1{p_2} = \frac1{q_1}+\frac1{q_2}$ .

The Hardy-Littlewood-Sobolev theorem implies the following $L^p$ type inequality:

Lemma 2.4. (^[16]) Let $0\leq s < \frac32$ , $1 < p\leq 2$ and $\frac12+\frac s3 = \frac1p$ , then

$\begin{equation} \|f\|_{\dot{H}^{-s}(\mathbb{R}^3)}\lesssim\|f\|_{L^p(\mathbb{R}^3)}. \end{equation}$

(2.5)

In the end, we introduce the special Sobolev interpolation lemma, which will be used in the proof of Theorem 1.4.

Lemma 2.5. (^[18]) Let $s\geq0$ and $l\geq0$ , then

$\begin{equation} \|\nabla^lf\|_{L^2(\mathbb{R}^3)}\leq\|\nabla^{l+1}f\|_{L^2(\mathbb{R}^3)}^{ 1-\theta }\|f\|_{\dot{H}^{-s}(\mathbb{R}^3)}^{ \theta },\quad{with}\; \theta = \frac 1{l+1+s}. \end{equation}$

(2.6)

3. Proof of Theorem 1.4

Equation (1.8) $_3$ can be rewritten as

$\begin{equation} (d-\omega_0)_t-\Delta(d-\omega_0) = -u\cdot\nabla (d-\omega_0)-[f(d)-f(\omega_0)]. \end{equation}$

(3.1)

In ^[19], the authors proved the $L^2$ -norm estimate of $d-\omega_0$ provided that the assumptions of Lemma 1.1 hold.

Lemma 3.1. (^[19]) Assume that all the assumptions of Lemma 1.1 hold. Then, the solution of (3.1) satisfies

$\begin{equation} \|d-\omega_0\|_{L^2}^2+\int_0^t\|\nabla (d-\omega_0)\|_{L^2}^2\leq \|d_0-\omega_0\|_{L^2}^2. \end{equation}$

(3.2)

In the following, we prove the decay estimates of strong solutions for system (1.8). The case $s\in[0, \frac12]$ was shown in Lemma 1.2, one only need to consider the case $s\in(\frac12, \frac32)$ . We first derive the evolution of the negative Sobolev norms of the solution.

Lemma 3.2. Under the assumptions of Lemma 1.1, if $s\in(\frac12, \frac32)$ , we have

$\begin{equation} \begin{aligned} & \frac d{dt}\int_{\mathbb{R}^3}(\gamma|\Lambda^{-s}\varrho|^2+|\Lambda^{-s}u|^2+|\Lambda^{-s}\nabla d|^2)dx +C\int_{\mathbb{R}^3}( |\nabla\Lambda^{-s}\nabla u|^2 +|\Lambda^{-s}\nabla^2d|^2)dx \\\leq&C \|\nabla d\|_{H^1}^2 (\|\Lambda^{-s}u\|_{L^2}+\|\Lambda^{-s}\varrho\|_{L^2}+\|\Lambda^{-s}\nabla d\|_{L^2}) \\ &+(\|\varrho\|_{L^2}+\|u\|_{L^2}+\|\nabla d\|_{L^2})^{s-\frac12}(\|\nabla\varrho\|_{H^1}+\|\nabla u\|_{H^1}+\|\nabla^2d\|_{H^1})^{\frac52-s}\\ &\times(\|\Lambda^{-s}u\|_{L^2}+\|\Lambda^{-s}\varrho\|_{L^2}+\|\Lambda^{-s}\nabla d\|_{L^2}). \end{aligned} \end{equation}$

(3.3)

Proof. Applying $\Lambda^{-s}$ to (1.8) $_1$ , (1.8) $_2$ , $\Lambda^{-s}\nabla$ to (1.8) $_3$ , multiplying the resulting identities by $\gamma\Lambda^{-s}\varrho$ , $\Lambda^{-s}u$ and $\Lambda^{-s}\nabla d$ respectively, summing up and integrating over $\mathbb{R}^3$ by parts, we arrive at

$\begin{equation} \begin{aligned} &\frac12\frac d{dt}\int_{\mathbb{R}^3}(\gamma|\Lambda^{-s}\varrho|^2+|\Lambda^{-s}u|^2+|\Lambda^{-s}\nabla d|^2)dx\\&+\int_{\mathbb{R}^3}(\bar{\mu}|\nabla\Lambda^{-s}u|^2+(\bar{\mu}|\nabla\Lambda^{-s}u|^2+(\bar{\mu}+\bar{\lambda})|\nabla\cdot\Lambda^{-s}u|^2+|\Lambda^{-s}\nabla^2d|^2)dx \\ = &\int_{\mathbb{R}^3}\gamma\Lambda^{-s}(-\varrho\nabla\cdot u-u\cdot\nabla\varrho)\cdot\Lambda^{-s}\varrho \\ &-\Lambda^{-s}[u\cdot\nabla u+h(\varrho) (\bar{\mu}\Delta u+(\bar{\mu}+\bar{\lambda})\nabla\nabla\cdot u)+g(\varrho)\nabla\varrho+\phi(\Delta d\cdot\nabla d)]\cdot\Lambda^{-s}u \\ &-\Lambda^{-s}\nabla(u\cdot\nabla d+f(d))\cdot\Lambda^{-s}\nabla ddx \\ = &K_1+K_2+K_3+K_4+K_5+K_6+K_7+K_8. \end{aligned} \end{equation}$

(3.4)

Note that $s\in(\frac12, \frac32)$ , it is easy to see that $\frac12+\frac s3 < 1$ and $\frac 3s\in(2, 6)$ . For the terms $K_1$ , by using Lemmas 2.2 and 2.4, Hölder's inequality, Young's inequality together with the estimates established in Lemma 1.1, we deduce that

$\begin{equation} \begin{aligned} K_1 = &-\int_{\mathbb{R}^3}\gamma\Lambda^{-s}(\varrho\nabla\cdot u)\cdot\Lambda^{-s}\varrho dx\leq C\|\Lambda^{-s}(\varrho\nabla\cdot u)\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\\ \leq &C\|\varrho\nabla\cdot u\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}\varrho\|_{L^2} \leq C\|\varrho\|_{L^{\frac 3s}}\|\nabla u\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2} \\ \leq&C\|\varrho\|_{L^2}^{s-\frac12}\|\nabla\varrho\|_{L^2}^{\frac32-s}\|\nabla u\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}, \end{aligned} \end{equation}$

(3.5)

similarly, for $K_2$ – $K_6$ , we have

$\begin{equation} \begin{aligned} K_2 = &-\int_{\mathbb{R}^3}\gamma\Lambda^{-s}(u\cdot\nabla\varrho)\cdot\Lambda^{-s}\varrho dx\leq C\|\Lambda^{-s}(u\cdot\nabla\varrho)\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\\ \leq &C\|u\cdot\nabla\varrho\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}\varrho\|_{L^2} \leq C\|u\|_{L^{\frac 3s}}\|\nabla \varrho\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\\\leq& C\|u\|_{L^2}^{s-\frac12}\|\nabla u\|_{L^2}^{\frac32-s}\|\nabla \varrho\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2},\end{aligned} \end{equation}$

(3.6)

$\begin{equation} \begin{aligned} K_3 = &-\int_{\mathbb{R}^3}\gamma\Lambda^{-s}(u\cdot\nabla u)\cdot\Lambda^{-s}\varrho dx\leq C\|\Lambda^{-s}(u\cdot\nabla u)\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\\ \leq &C\|u\cdot\nabla u\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}\varrho\|_{L^2} \leq C\|u\|_{L^{\frac 3s}}\|\nabla u\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\\\leq&C\|u\|_{L^2}^{s-\frac12}\|\nabla u\|_{L^2}^{\frac32-s}\|\nabla u\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}, \end{aligned} \end{equation}$

(3.7)

$\begin{equation} \begin{aligned} K_4 = &-\int_{\mathbb{R}^3} \Lambda^{-s}[h(\varrho) (\bar{\mu}\Delta u+(\bar{\mu}+\bar{\lambda})\nabla\nabla\cdot u)]\cdot\Lambda^{-s}\varrho dx \\ \leq&\|\Lambda^{-s}[h(\varrho) (\bar{\mu}\Delta u+(\bar{\mu}+\bar{\lambda})\nabla\nabla\cdot u)]\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2} \\ \leq&C\|h(\varrho) (\bar{\mu}\Delta u+(\bar{\mu}+\bar{\lambda})\nabla\nabla\cdot u)\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}\varrho\|_{L^2} \\ \leq&C \|h(\varrho)\|_{L^{\frac 3s}}\|\nabla^2u\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\leq C\|\varrho\|_{L^{\frac 3s}}\|\nabla^2u\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2} \\\leq & C\|\varrho\|_{L^2}^{s-\frac12}\|\nabla \varrho\|_{L^2}^{\frac32-s}\|\nabla^2u\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2},\end{aligned} \end{equation}$

(3.8)

$\begin{equation} \begin{aligned} K_5 = &-\int_{\mathbb{R}^3} \Lambda^{-s}[g(\varrho)\nabla\varrho]\cdot\Lambda^{-s}\varrho dx\leq\|\Lambda^{-s}[g(\varrho)\nabla\varrho]\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2} \\ \leq&C\|g(\varrho)\nabla\varrho\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}\varrho\|_{L^2} \\ \leq&C \|g(\varrho)\|_{L^{\frac 3s}}\|\nabla\varrho\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\leq C\|\varrho\|_{L^{\frac 3s}}\|\nabla\varrho\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}\\ \leq &C\|\varrho\|_{L^2}^{s-\frac12}\|\nabla \varrho\|_{L^2}^{\frac32-s}\|\nabla \varrho\|_{L^2}\|\Lambda^{-s}\varrho\|_{L^2}, \end{aligned} \end{equation}$

(3.9)

and

$\begin{equation} \begin{aligned} K_6 = &-\int_{\mathbb{R}^3}\Lambda^{-s}(\phi(\varrho)\nabla d\cdot\Delta d)\cdot\Lambda^{-s}udx \\ \leq&C\|\Lambda^{-s}(\phi(\varrho)\nabla d\cdot\Delta d)\|_{L^2}\| \Lambda^{-s}u\|_{L^2} \\ \leq&C\|\phi(\varrho)\nabla d\cdot\Delta d\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}u\|_{L^2}\\ \leq&C\|\phi(\varrho)\|_{L^\infty}\|\nabla d\cdot\Delta d\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}u\|_{L^2} \\ \leq&C\|\nabla d\cdot\Delta d\|_{L^{\frac1{\frac12+\frac s3}}}\|\Lambda^{-s}u\|_{L^2} \leq C\|\nabla d\|_{L^{\frac 3s}}\|\Delta d\|_{L^2}\|\Lambda^{-s}u\|_{L^2} \\ \leq&C\|\nabla d\|_{L^2}^{s-\frac12}\|\Delta d\|_{L^2}^{\frac32-s}\|\Delta d\|_{L^2}\|\Lambda^{-s}u\|_{L^2}, \end{aligned} \end{equation}$

(3.10)

where we have used the fact (1.9) in (3.8)–(3.10). Next, by using Lemmas 2.2–2.4, Hölder's inequality, Young's inequality together with Lemma 1.1 on the energy estimates of the solutions, it yields that

$\begin{equation} \begin{aligned} K_7 = &-\int_{\mathbb{R}^3}\Lambda^{-s}\nabla(u\cdot\nabla d)\cdot\Lambda^{-s}\nabla ddx \\ \leq&C\|\Lambda^{-s}(\nabla u\cdot\nabla d+u\cdot\nabla^2d)\|_{L^2}\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C(\|\nabla u\cdot\nabla d\|_{L^{\frac1{\frac12+\frac s3}}}+\|u\cdot\nabla^2d\|_{L^{\frac1{\frac12+\frac s3}}})\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C(\|\nabla d\|_{L^{\frac 3s}}\|\nabla u\|_{L^2}+\|u\|_{L^{\frac 3s}}\|\nabla^2d\|_{L^2})\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C(\|\nabla d\|_{L^2}^{s-\frac12}\|\nabla^2d\|_{L^2}^{\frac32-s}\|\nabla u\|_{L^2}+\|u\|_{L^2}^{s-\frac12}\|\nabla u\|_{L^2}^{\frac32-s}\|\nabla^2d\|_{L^2})\|\Lambda^{-s}\nabla d\|_{L^2}.\end{aligned} \end{equation}$

(3.11)

For $K_8$ , we first consider $s\in(\frac12, 1)$ . Thanks to Lemma 2.2, one easily obtain

$\|\Lambda^{2-s}d\|_{L^2}+\|\Lambda^{2-s}(d-\omega_0)\|_{L^2}+\|\Lambda^{2-s}(d+\omega_0)\|_{L^2}\leq C\|\nabla d\|_{L^2}^s\|\nabla^2d\|_{L^2}^{1-s}\leq C(\|\nabla d\|_{L^2}+\|\nabla^2d\|_{L^2}).$

Then, by Hölder's inequality, Young's inequality, the facts $|d| < 1$ , $|\omega_0| = 1$ together with Lemmas 1.1, 2.2 and 2.3, we derive that

$\begin{equation} \begin{aligned} K_8 = &-\int_{\mathbb{R}^3}\Lambda^{-s}\nabla\cdot[(d+\omega_0)(d-\omega_0)d]\cdot\Lambda^{-s}\nabla ddx \\ \leq&C\|\Lambda^{-s}\nabla\cdot[(d+\omega_0)(d-\omega_0)d]\|_{L^2}\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C[\|d+\omega_0\|_{L^6}\|d-\omega_0\|_{L^6}\|\Lambda^{1-s}d\|_{L^6}+\|d+\omega_0\|_{L^6}\|d\|_{L^6}\|\Lambda^{1-s}(d-\omega_0)\|_{L^6}\\&+\|d-\omega_0\|_{L^6}\|d \|_{L^6}\|\Lambda^{1-s}(d+\omega_0)\|_{L^6}]\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C\|\nabla d\|_{L^2}^2(\|\Lambda^{2-s}d\|_{L^2}+\|\Lambda^{2-s}(d-\omega_0)\|_{L^2}+\|\Lambda^{2-s}(d+\omega_0)\|_{L^2})\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C\|\nabla d\|_{L^2} (\|\nabla d\|_{L^2}+\|\nabla^2d\|_{L^2})\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C(\|\nabla d\|_{L^2}^2+\|\nabla^2d\|_{L^2}^2)\|\Lambda^{-s}\nabla d\|_{L^2}. \end{aligned} \end{equation}$

(3.12)

Moreover, if $s\in(1, \frac32)$ , the following inequality holds:

$\begin{equation} \begin{aligned} K_8 \leq&C\|\Lambda^{-s+1}[(d+\omega_0)(d-\omega_0)d]\|_{L^2}\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C \|(d+\omega_0)(d-\omega_0) d\|_{L^{\frac1{\frac12+\frac{s-1}3}}}\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C \|d+\omega_0\|_{L^{\infty}}\|d-\omega_0\|_{L^2}\|d\|_{L^{\frac 3{s-1}}}\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C\|\nabla d\|_{L^2}^{\frac12}\|\nabla^2d\|_{L^2}^{\frac12}\|d-\omega_0\|_{L^2}\|\nabla d\|_{L^2}^{(s-1)+\frac12}\|\nabla^2d\|_{L^2}^{\frac12-(s-1)}\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C\|\nabla d\|_{L^2}^s\|\nabla^2d\|_{L^2}^{2-s}\|\Lambda^{-s}\nabla d\|_{L^2} \\ \leq&C(\|\nabla d\|_{L^2}^2+\|\nabla^2d\|_{L^2}^2)\|\Lambda^{-s}\nabla d\|_{L^2}. \end{aligned} \end{equation}$

(3.13)

Combining (3.4)–(3.12) together, we obtain (3.3) and complete the proof.

Now, we give the proof of our main results.

Proof of Theorem 1.4. First of all, the sketch of proof for the decay estimate with $s\in[0, \frac12]$ will be derived in the following. Note that this part follows more or less the lines of ^[19], so that we do note claim originality here. Then, by using this proved estimate, one can obtain the decay results for $s\in(\frac12, \frac32)$ .

Now, consider the decay for $s\in[0, \frac12]$ . We first establish the negative Sobolev norm estimates for the strong solutions, obtain one important inequality:

$\begin{equation} \begin{aligned}\nonumber& \frac d{dt}(\gamma\|\Lambda^{-s}\varrho\|_{L^2}^2+\|\Lambda^{-s}u\|_{L^2}^2+\|\Lambda^{-s}\nabla d\|_{L^2}^2) +C(\|\Lambda^{-s}\nabla u\|_{L^2}^2+\|\Lambda^{-s}\nabla^2d\|_{L^2}^2) \\ \lesssim&(\|\nabla(\varrho,u)\|_{H^1}^2+\|\nabla d\|_{H^2}^2)(\|\Lambda^{-s}\varrho\|_{L^2}^2+\|\Lambda^{-s}u\|_{L^2}^2+\|\Lambda^{-s}\nabla d\|_{L^2}^2). \end{aligned} \end{equation}$

Then, define

$\mathcal{E}_{-s}(t) = \|\Lambda^{-s}\varrho(t)\|_{L^2}^2+\|\Lambda^{-s}u(t)\|_{L^2}^2+\|\Lambda^{-s}\nabla d(t)\|_{L^2}^2,$

we deduce from (3.2) and (3.3) that for $s\in[0, \frac12]$ ,

$\mathcal{E}_{-s}(t)\leq\mathcal{E}_{-s}(0)+C\int_0^t(\|\nabla(\varrho,u)\|_{H^1}^2+\|\nabla d\|_{H^2}^2)\sqrt{\mathcal{E}_{-s}(t)}d\tau\leq C(1+\sup\limits_{0\leq\tau\leq t}\sqrt{\mathcal{E}_{-s}(t)}),$

which implies (1.12) for $s\in[0, \frac12]$ , i.e.,

$\begin{equation} \|\Lambda^{-s}\varrho(t)\|_{L^2}^2+\|\Lambda^{-s}u(t)\|_{L^2}^2+\|\Lambda^{-s}\nabla d(t)\|_{L^2}^2 \leq C_0. \end{equation}$

(3.14)

Moreover, if $l = 1, 2, \cdots, N-1$ , we may use Lemma 2.4 to have

$\begin{equation} \|\nabla^{l+1}f\|_{L^2}\geq C\|\Lambda^{-s}f\|_{L^2}^{-\frac1{l+s}}\|\nabla^lf\|_{L^2}^{1+\frac1{l+s}}. \end{equation}$

(3.15)

Then, by (3.14) and (3.15), it yields that

$\begin{equation*} \label{7-1r} \|\nabla^{l}(\nabla \varrho, \nabla u,\nabla^2d)\|_{L^2}^2\geq C (\|\nabla^{l}(\varrho,u, \nabla d)\|_{L^2}^2)^{1+\frac1{l+s}}. \end{equation*}$

Hence, for $l = 1, 2, \cdots, N-1$ ,

$\begin{equation} \|\nabla^{l}(\nabla \varrho, \nabla u,\nabla^2d)\|_{H^{N-l-1}}^2\geq C (\|\nabla^{l}(\varrho,u, \nabla d)\|_{H^{N-l}}^2)^{1+\frac1{l+s}}. \end{equation}$

(3.16)

Thus, we deduce from (1.11) the following inequality

$\begin{equation} \nonumber\begin{aligned}& \frac d{dt}(\|\nabla^l\varrho\|_{H^{N-l}}^2+\|\nabla^lu\|_{H^{N-l}}^2 +\| \nabla^{l+1} d \|_{H^{N-l}}^2)\\&+C_0(\|\nabla^l\varrho\|_{H^{N-l}}^2+\|\nabla^lu\|_{H^{N-l}}^2 +\| \nabla^{l+1} d \|_{H^{N-l}}^2)^{1+\frac1{l+s}}\leq 0,\; \hbox{for}\; l = 1, \cdots,N-1. \end{aligned} \end{equation}$

Solving this inequality directly gives

$\begin{equation} \|\nabla^l\varrho\|_{H^{N-l}}+\|\nabla^lu\|_{H^{N-l}}+\|\nabla^{l+1}d\|_{H^{N-l}}\leq C (1+t)^{-\frac{l+s}2},\quad\hbox{for}\; l = 1,\cdots,N-1. \end{equation}$

(3.17)

Then, by (3.14), (3.17) and the interpolation, we obtain the following inequality holds for $s\in[0, \frac12]$ :

$\begin{equation} \|\nabla^l\varrho\|_{H^{N-l}}+\|\nabla^lu\|_{H^{N-l}}+\|\nabla^{l+1}d\|_{H^{N-l}}\leq C (1+t)^{-\frac{l+s}2},\quad\hbox{for}\; l = 0, 1,\cdots,N-1. \end{equation}$

(3.18)

Second, we consider the decay estimate for $s\in(\frac12, \frac32)$ . Notice that the arguments for $s\in[0, \frac12]$ can not be applied to this case. However, observing that we have $\varrho_0, u_0, \nabla d_0\in\dot{H}^{-\frac12}$ hold since $\dot{H}^{-s}\bigcap L^2\subset\dot{H}^{-s'}$ for any $s'\in[0, s]$ , we can deduce from (3.18) for (1.10) and (1.11) with $s = \frac12$ that the following estimate holds:

$\begin{equation} \|\nabla^l\varrho\|_{H^{N-l}}^2+\|\nabla^lu\|_{H^{N-l}}^2+\|\nabla^l\nabla d\|_{H^{N-l}}^2 \leq C_0(1+t)^{-\frac12-l},\quad \hbox{for}\; l = 0,1,\cdots,N-1. \end{equation}$

(3.19)

Therefore, we deduce from (3.3) and (3.2) that for $s\in(\frac12, \frac32)$ ,

$\begin{equation} \begin{aligned} & \mathcal{E}_{-s}(t)\\\leq&\mathcal{E}_{-s}(0)+C\int_0^t\|\nabla d\|_{H^1}^2 \sqrt{\mathcal{E}_{-s}(\tau)}d\tau \\&+C\int_0^t (\|\varrho\|_{L^2}+\|u\|_{L^2}+\|\nabla d\|_{L^2})^{s-\frac12}(\|\nabla\varrho\|_{H^1}+\|\nabla u\|_{H^1}+\|\nabla^2d\|_{H^1})^{\frac52-s} \sqrt{\mathcal{E}_{-s}(\tau)}d\tau \\ \leq&C_0+C \sup\limits_{\tau\in[0,t]} \sqrt{\mathcal{E}_{-s}(\tau)}+C\int_0^t(1+\tau)^{-\frac74+\frac s2}d\tau\sup\limits_{\tau\in[0,t]} \sqrt{\mathcal{E}_{-s}(\tau)} \\ \leq&C_0+C\sup\limits_{\tau\in[0,t]} \sqrt{\mathcal{E}_{-s}(\tau)}, \end{aligned} \end{equation}$

(3.20)

which implies that (1.12) holds for $s\in(\frac12, \frac32)$ , i.e.,

$\begin{equation} \|\Lambda^{-s}\varrho(t)\|_{L^2}^2+\|\Lambda^{-s}u(t)\|_{L^2}^2+\|\Lambda^{-s}\nabla d(t)\|_{L^2}^2 \leq C_0. \end{equation}$

(3.21)

Moreover, thanks to (1.11) and (3.16), we can also obtain the following inequality for $s\in(\frac12, \frac32)$ :

which implies

$\begin{equation} \|\nabla^l\varrho\|_{H^{N-l}}+\|\nabla^lu\|_{H^{N-l}}+\|\nabla^{l+1}d\|_{H^{N-l}}\leq C (1+t)^{-\frac{l+s}2},\quad\hbox{for}\; l = 1,\cdots,N-1. \end{equation}$

(3.22)

Next, using (3.21), (3.22), and Lemma 2.5, we easily obtain

$\begin{equation} \begin{aligned} \|(\varrho,u,\nabla d)\|_{L^2}\leq& C(\|\nabla(\varrho,u,\nabla d)\|_{L^2})^{\frac s{1+s}}(\|\Lambda^{-s}(\varrho,u,\nabla d)\|_{L^2}^{\frac1{1+s}} \\ \leq&C(\|\nabla(\varrho,u,\nabla d)\|_{L^2})^{\frac s{1+s}} \\ \leq&C\left[(1+t)^{-\frac{1+s}2}\right]^{\frac s{1+s}} = C(1+t)^{-\frac{s}2}. \end{aligned} \end{equation}$

(3.23)

It then follows from (3.22) and (3.23) that

$\|\varrho\|_{H^{N-l}}+\| u\|_{H^{N-l}}+\|\nabla d\|_{H^{N-l}}\leq C(1+t)^{-\frac{s}2}.$

Hence, we obtain (1.15) for $s\in(\frac12, \frac32)$ and complete the proof.

4. Conclusions

In this paper, we consider the optimal decay estimates for the higher order derivatives of strong solutions for three-dimensional nematic liquid crystal system. We use the pure energy method, negative Sobolev norm estimates together with the classical Kato-Ponce inequality, Gagliardo-Nirenberg inequality, overcome the difficulties caused by the Ginzburg-Landau approximation and the coupling between the compressible Navier-Stokes equations and the direction equations, obtain the decay estimates. Since the result (1.16) is same to the decay of the heat equation, it is optimal. We remark that our results may attract the attentions of the researchers in the nematic liquid crystals filed.

Acknowledgments

The author would like to thank the anonymous referees and Dr. Xiaopeng Zhao for their helpful suggestions. This paper was supported by the Fundamental Research Funds of Heilongjiang Province (grant No. 145109131).

Conflict of interest

The author declares no conflict of interest.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Seegers WH, Johnson JF, Fell C (1954) An antthrombin reaction to prothrombin activation. Am J Physiol 176: 97-103. https://doi.org/10.1152/ajplegacy.1953.176.1.97
[2]	Egeberg O (1965) Inherited antithrombin deficiency causing thrombophilia. Thromb Diath Haemorrh 13: 516-530. https://doi.org/10.1055/s-0038-1656297
[3]	Griffin JH, Evatt B, Zimmerman TS, et al. (1981) Deficiency of protein C in congenital thrombotic disease. J Clin Invest 68: 1370-1373. https://doi.org/10.1172/JCI110385
[4]	Comp PC, Esmon C (1984) Recurrent venous thromboembolism in patients with a partial deficiency of protein S. N Engl J Med 311: 1525-1528. https://doi.org/10.1056/NEJM198412133112401
[5]	Schwarz HP, Fischer M, Hopmeier P, et al. (1984) Plasma protein S deficiency in familial thrombotic disease. Blood 64: 1297-1300. https://doi.org/10.1182/blood.V64.6.1297.1297
[6]	Conard J, Brosstad F, Lie Larsen M, et al. (1983) Molar antithrombin concentration in normal human plasma. Haemostasis 13: 363-368. https://doi.org/10.1159/000214823
[7]	Collen D, Schetz J, de Cock F, et al. (1977) Holmer, E., Verstraete, M. Metabolism of antithrombin III (heparin cofactor) in man: effects of venous thrombosis of heparin administration. Eur J Clin Invest 7: 27-35. https://doi.org/10.1111/j.1365-2362.1977.tb01566.x
[8]	Maclean PS, Tait RC (2007) Hereditary and acquired antithrombin deficiency: epidemiology, pathogenesis and treatment options. Drugs 67: 1429-1440. https://doi.org/10.2165/00003495-200767100-00005
[9]	Wells PS, Blajchman MA, Henderson P, et al. (1994) Prevalence of antithrombin deficiency in healthy blood donors: a cross-sectional study. Am J Hematol 45: 321-324. https://doi.org/10.1002/ajh.2830450409
[10]	Tait RC, Walker ID, Perry DJ, et al. (1994) Prevalence of antithrombin deficiency in the healthy population. Br J Haematol 87: 106-112. https://doi.org/10.1111/j.1365-2141.1994.tb04878.x
[11]	Picard V, Nowark-Gottl U, Biron-Andreani C, et al. (2006) Molecular bases of antithrombin deficiency: twenty-two novel mutations in the antithrombin gene. Hum Mutat 27: 600. https://doi.org/10.1002/humu.9425
[12]	Khor B, Van Cott EM (2010) Laboratory tests for antithrombin deficiency. Am J Hematol 85: 947-950. https://doi.org/10.1002/ajh.21893
[13]	Patnaik MM, Moll S (2008) Inherited antithrombin deficiency: a review. Haemophilia 14: 1229-1239. https://doi.org/10.1111/j.1365-2516.2008.01830.x
[14]	Dahlback B (2008) Advances in understanding pathogenic mechanisms of thrombophilic disorders. Blood 112: 19-27. https://doi.org/10.1182/blood-2008-01-077909
[15]	Rodgers GM (2009) Role of antithrombin concentrate in treatment of hereditary antithrombin deficiency. An update. Thromb Haemost 101: 806-812. https://doi.org/10.1160/TH08-10-0672
[16]	Nguyen KV (2012) Antithrombin Hanoi: Arg393 to His missense point mutation in antithrombin gene and cancer. WebmedCentral: International Journal of Medicine and Molecular Medicine 3: WMC003720.
[17]	Bock SC (2001) Anthithrombin III and heparin cofactor II. Hemostasis and Thrombosis: Basic Principles and Clinical Practice . Philadelphia, Pa: Lippincott Williams & Wilkins pp. 321-333.
[18]	Walker CPR, Royston D (2002) Thrombin generation and its inhibition: a review of the scientific basis and mechanism of action of anticoagulant therapies. Br J Anaesth 88: 848-863. https://doi.org/10.1093/bja/88.6.848
[19]	Morawitz P (1905) Die Chemie der Blutgerinnung. Ergeb Physiol 4: 307-319. https://doi.org/10.1007/BF02321003
[20]	Quick AJ (1938) The normal antithrombin of the blood and its relation to heparin. Am J Physiol 123: 712-719. https://doi.org/10.1152/ajplegacy.1938.123.3.712
[21]	Fell C, Ivanovic N, Johnson SA, et al. (1954) Differenciation of plasma antithrombin activities. Proc Soc Exp Biol Med 85: 199-202. https://doi.org/10.3181/00379727-85-20829
[22]	Sakuragawa N (1997) Regulation of thrombosis and hemostasis by antithrombin. Semin Thromb Hemost 23: 557-562. https://doi.org/10.1055/s-2007-996136
[23]	Mammen EF (1998) Antithrombin: its physiological importance and role in DIC. Semin Thromb Hemost 24: 19-25. https://doi.org/10.1055/s-2007-995819
[24]	Bock SC, Harris JF, Balazs I, et al. (1985) Assignment of the human antithrombin III structural gene to chromosome 1q23-25. Cytogenet Cell Genet 39: 67-69. https://doi.org/10.1159/000132105
[25]	Olds RJ, Lane DA, Chowdhury V, et al. (1993) Complete nucleotide sequence of the antithrombin gene: evidence for homologous recombination causing thrombophilia. Biochemistry 32: 4216-4224. https://doi.org/10.1021/bi00067a008
[26]	Van Boven HH, Lane DA (1997) Antithrombin and its inherited deficiency states. Semin Hematol 34: 188-204.
[27]	Bock SC, Marriman JA, Radziejewska E (1998) Antithrombin III Utah: proline- 407 to leucine mutation in a highly conserved region near the inhibitor reactive site. Biochemistry 27: 6171-6178. https://doi.org/10.1021/bi00416a052
[28]	Petersen TE, Dulek-Wojciechowski G, Sottrup-Jensen L, et al. (1979) Primary structure of antithrombin III (heparin cofactor): partial homology between alpha 1 antitrypsin and antithrombin III. The physiology inhibitors of blood coagulation and fibrinolysis . Amsterdam, The Netherlands: Elsevier Science pp. 43-54.
[29]	Bock SC, Wion KL, Vehar GA, et al. (1982) Cloning and expression of the cDNA for human antithrombin III. Nucleic Acids Res 10: 8113-8125. https://doi.org/10.1093/nar/10.24.8113
[30]	Prochownik EV, Orkin SH (1984) In vivo transcription of a human antithrombin III “minigene”. J Biol Chem 259: 15386-15392. https://doi.org/10.1016/S0021-9258(17)42561-0
[31]	Patston PA, Gettins P, Beechem J, et al. (1991) Mechanism of serpin action: evidence thet C1 inhibitor functions as a suicide substrate. Biochemistry 30: 8876-8882. https://doi.org/10.1021/bi00100a022
[32]	Bjork J, Jackson CM, Jornvall H, et al. (1982) The active site of antithrombin. Release of the same proteolytically cleaved form of the inhibitor from complexes with factor IXa, factor Xa, and thrombin. J Biol Chem 257: 2406-2411. https://doi.org/10.1016/S0021-9258(18)34938-X
[33]	Marchant KK, Ducan A (2002) Antithrombin deficiency. Issue in laboratory diagnosis. Arch Pathol Lab Med 126: 1326-1336. https://doi.org/10.5858/2002-126-1326-AD
[34]	Perry DJ (1994) Antithrombin and its inherited deficiencies. Blood Rev 8: 37-55. https://doi.org/10.1016/0268-960X(94)90006-X
[35]	Lane DA, Olds RJ, Thein SL (1994) Antithrombin and its deficiency. Hemostasis and thrombosis . Edinburgh: Churchils Livingstone pp. 655-670.
[36]	Hirsh J, Piovella F, Pini M (1989) Congenital antithrombin III deficiency: incidence and clinical features. Am J Med 87: 34S-38S. https://doi.org/10.1016/0002-9343(89)80529-7
[37]	Koide T, Odani S, Takahashi K, et al. (1984) Antithrombin III Toyama: replacement of arginine-47 by cysteine in hereditary abnormal antithrombin III that lacks heparin-binding ability. Proc Natl Acad Sci USA 81: 289-293. https://doi.org/10.1073/pnas.81.2.289
[38]	Chang JY, Tran TH (1986) Antithrombin III Basel. Identification of a pro-leu substitution in a hereditary abnormal antithrombin with impaired heparin cofactor activity. J Biol Chem 261: 1174-1176. https://doi.org/10.1016/S0021-9258(17)36071-4
[39]	Owen MC, Borg JY, Soria C, et al. (1987) Heparin binding defect in a new antithrombin III variant: Rouen, 47 Arg to His. Blood 69: 1275-1279. https://doi.org/10.1182/blood.V69.5.1275.1275
[40]	Stephens AW, Thalley BS, Hirs CHW (1987) Antithrombin-III Denver, a reactive site variant. J Biol Chem 262: 1044-1048. https://doi.org/10.1016/S0021-9258(19)75747-0
[41]	Lane DA, Flynn A, Ireland H, et al. (1987) Antithrombin III Nortwick Park: demonstration of an inactive high MW complex with increased affinity for heparin. Br J Haematol 65: 451-456. https://doi.org/10.1111/j.1365-2141.1987.tb04149.x
[42]	Erdjument H, Lane DA, Ireland H, et al. (1987) Formation of a covalent disulfide- linked antithrombin-albumin complex by an antithrombin variant, antithrombin “Northwick Park”. J Biol Chem 262: 13381-13384. https://doi.org/10.1016/S0021-9258(19)76436-9
[43]	Lane DA, Lowe GDO, Flynn A, et al. (1987) Antithrombin III Glasgow: a variant with increased heparin affinity and reduced ability to inactivate thrombin, associated with familial thrombosis. Br J Haematol 66: 523-527. https://doi.org/10.1111/j.1365-2141.1987.tb01338.x
[44]	Owen MC, Beresford CH, Carrell RW (1988) Antithrombin Glasgow, 393 Arg to His: a P1 reactive site variant with increased heparin affinity but no thrombin inhibitory activity. FEBS Lett 231: 317-320. https://doi.org/10.1016/0014-5793(88)80841-X
[45]	Lane DA, Erdjument AF, Flynn A, et al. (1989) Antithrombin Sheffield: amino acid substitution at the reactive site (Arg393 to His) causing thrombosis. Br J Haematol 71: 91-96. https://doi.org/10.1111/j.1365-2141.1989.tb06280.x
[46]	Kenji O, Hiroki A, Masako W, et al. (1995) Antithrombin III Kumamoto: a single mutation at Arg393-His increases the affinity of Antithrombin III for heparin. Am J Hematol 48: 12-18. https://doi.org/10.1002/ajh.2830480104
[47]	Ulivi L, Squitieri M, Cohen H, et al. (2020) Cerebral venous thrombosis: a practical guide. Pract Neurol 20: 356-367. https://doi.org/10.1136/practneurol-2019-002415
[48]	Betts MJ, Russell RB (2003) Amino acid properties and consequences of substitutions. Bioinformatics for geneticists . John Wiley & Sons, Ltd pp. 289-316. https://doi.org/10.1002/0470867302.ch14
[49]	Kyrle PA, Minar E, Hirschl M, et al. (2000) High plasma levels of Factor VIII and the risk of recurrent venous thromboembolism. N Engl J Med 343: 457-462. https://doi.org/10.1056/NEJM200008173430702
[50]	Kamphuisen PW, Eikenboom JCJ, Bertina RM (2001) Elevated factor VIII levels and the risk of thrombosis. Arterioscler Thromb Vasc Biol 21: 731-738. https://doi.org/10.1161/01.ATV.21.5.731
[51]	Garcia JH, Williams JP, Tanaka J (1975) Spontaneous thrombosis of deep cerebral veins: a complication of arteriovenous malformation. Stroke 6: 164-171. https://doi.org/10.1161/01.STR.6.2.164
[52]	Shah Sid “Stroke pathophysiology.” Foundation for education and research in neurological emergencies. [FERNE]: 1–14. web (2013). Available from: http://tigger.uic.edu/com/ferne/pdf2/saem_0501/shah_stroke_0501.pdf
[53]	Ord-Mackenzi SA (1898) Transient and recurring paresis in acute cerebral softening. Br Med J 1: 140. https://doi.org/10.1136/bmj.1.1933.140
[54]	Fourrier F, Lestavel P, Chopin C, et al. (1990) Meningococcemia and purpura fulminants in adults: acute deficiencies of proteins C and S and early treatment with antithrombin III concentrates. Intensive Care Med 16: 121-124. https://doi.org/10.1007/BF02575306
[55]	Bozzola E, Bozzola M, Colafati GS, et al. (2014) Multiple cerebral sinus thromboses complicating meningococcal meningitis: a pediatric case report. BMC Pediatrics 14: 147. https://doi.org/10.1186/1471-2431-14-147
[56]	Fawcett JW, Asher RA (1999) The glial scar and central nervous system repair. Brain Res Bull 49: 377-391. https://doi.org/10.1016/S0361-9230(99)00072-6
[57]	Heparin antifactor Xa assay (2009). (http://www2.massgeneral.org/pathology/coagbook/CO005000.htm) Archived (https://archive.is/20090808194520/ http://www2.massgeneral.org/pathology/coagbook/CO005000.htm) 2009- 08-08 at Archive.today Page Updated: September 18, 2009
[58]	Refael M, Xing L, Lim W, et al. (2017) Management of venous thromboembolism in patients with hereditary antithrombin deficiency and pregnancy: case report and review of the literature. Case Rep Hematol : Article ID: 9261351. https://doi.org/10.1155/2017/9261351
[59]	Ansell J, Hirsh, Hylek E, et al. (2008) Pharmacology and management of the vitamin K antagonists: American college Chest physicians evidence-based clinical practice guidelines (8^th edition). Chest 133: 160S-198S. https://doi.org/10.1378/chest.08-0670
[60]	Ageno W, Gallus AS, Wittkowsky A, et al. (2012) Oral anticoagulant therapy: antithrombotic therapy and prevention of thrombosis, 9^th ed.: American college Chest physicians evidence-based clinical 141: practice guidelines. Chest 141: e44S-e88S. https://doi.org/10.1378/chest.11-2292
[61]	. Coumadin (https//www.drugs.com/monograph/coumadin.htlm). The American society of health-system pharmacists. Archived (https//web.archive.org/web/20110203081242/ http://www.drugs.com/monograph/coumadin.html) from the original on 3 February 2011. Retrieved 3 April 2011.
[62]	. Warfarin sodium (https://www.drugs.com/monograph/warfarin-sodium.html). The American society of health-system pharmacists. Archived (https://web.archive.org/web/20180612143838/ https://www.drugs.com/monograph/warfarin-sodium.html) from the original on 12 June 2018. Retrieved 8 January 2017.
[63]	Weathermor R, Crabb DW (1999) Alcohol and medication interactions. Alcohol Res Health 23: 40-54.
[64]	Clinically significant drug interactions - American family physician (2016). (http://www.aafp.org/afp/2000/0315/p1745.html#sec-1) Archived (https://web.archive.org/web/20160507065339/http://www.aafp.org/afp/2000/0315/p1745.html) 7 May 2016 at the Wayback Machine
[65]	Wang KL, Yap ES, Goto S, et al. (2018) The diagnosis and treatment of venous thromboembolism in Asian patients. Thromb J 16: 4. https://doi.org/10.1186/s12959-017-0155-z
[66]	Darvall KAL, Sam RC, Silverman SH, et al. (2007) Obesity and thrombosis. Eur J Vasc Surg 33: 223-233. https://doi.org/10.1016/j.ejvs.2006.10.006
[67]	Ridker PM, Miletich JP, Hennekens CH, et al. (1997) Ethnic distribution of factor V Leiden in 4047 men and women. Implications for venous thromboembolism screening. JAMA 277: 1305-1307. https://doi.org/10.1001/jama.1997.03540400055031
[68]	Varga EA, Moll S (2004) Prothrombin 20210 mutation (factor II mutation). Circulation 110: e15-e18. https://doi.org/10.1161/01.CIR.0000135582.53444.87
[69]	Weinberger J, Cipolle M (2016) Mechanism prophylaxix for post-traumatic VTE: stockings and pumps. Curr Trauma Rep 2: 35-41. https://doi.org/10.1007/s40719-016-0039-x
[70]	Liew NC, Chang YH, Choi G, et al. (2012) Asian venous thrombosis forum. Asian venous thromboembolism guidelines: prevention of venous thromboembolism. Int Angiol 31: 501-516.
[71]	ISTH Steering Committee for World Thrombosis Day.Thrombosis: a major contributor to global disease burden. Thromb Haemost (2014) 112: 843-852. https://doi.org/10.1160/th14-08-0671
[72]	Hotoleanu C (2020) Association between obesity and venous thromboembolism. Med Pharm Rep 93: 162-168. https://doi.org/10.15386/mpr-1372
[73]	Aksu K, Donmez A, Keser G (2012) Inflammation-induced thrombosis: mechanism, disease associations management. Curr Pharma Des 18: 1478-1493. https://doi.org/10.2174/138161212799504731
[74]	Middellorp S, Coppens M, van Happs F (2020) Incidence of venous thromboembolism in hospitalized patients with COVID-19. J Thromb Haemost 18: 1995-2002. https://doi.org/10.1111/jth.14888
[75]	Connors JM, Levy JH (2020) COVID-19 and its implications for thrombosis and anticoagulation. Blood 135: 2033-2040. https://doi.org/10.1182/blood.2020006000
[76]	Vinogradova Y, Coupland C, Hippisley-Cox J (2019) Use of hormone replacement therapy and risk of venous thromboembolism: nested case-control studies using the QResearch and CPRD databases. BMJ 364: k4810. https://doi.org/10.1136/bmj.k4810
[77]	Ye F, Bell LN, Mazza J, et al. (2018) Variation in definitions of immobility in pharmacological thromboprophylaxis clinical trials in medical inpatients: a systemic review. Clin Appl Thromb Hemost 24: 13-21. https://doi.org/10.1177/1076029616677802
[78]	Fernandes CJ, Morinaga LTK, Alves JL, et al. (2019) Cancer-associated thrombosis: the when, how and why. Eur Respir Rev 28: 180119. https://doi.org/10.1183/16000617.0119-2018
[79]	Noble S, Pasi J (2010) Epidemiology and pathophysiology of cancer-associated thrombosis. Br J Cancer 102: S2-S9. https://doi.org/10.1038/sj.bjc.6605599
[80]	Bouillaud JB, Bouillaud S (1823) De l'Obliteration des veines et de son influence sur la formation des hydropisies partielles: consideration sur la hydropisies passive et general. Arch Gen Med 1: 188-204.
[81]	Trousseau A (1865) Phlegmasia alba dolens. Clinique medicale de l'Hotel- Dieu de Paris . Paris: Balliere pp. 654-712.
[82]	Trousseau A (1877) Ulcere chronique simple de l'estomac. Clinique medicale de l'Hotel-Dieu de Paris . Paris: Bailliere J.-B. et fils pp. 80-107.
[83]	Trousseau A (1877) Phlegmasia alba dolens. Clinique medicale de l'Hotel-Dieu de Paris . Paris: Bailliere J.-B. et fils pp. 695-739.
[84]	Illtyd James T, Matheson N (1935) Thromboplebitis in cancer. Practitioner 134: 683-684.
[85]	Gore JM, Appelbaum JS, Greene HL, et al. (1982) Occult cancer in patients with acute pulmonary embolism. Ann Intern Med 96: 556-560. https://doi.org/10.7326/0003-4819-96-5-556
[86]	Prandoni P, Lensing AWA, Buller HR, et al. (1992) Deep-vein thrombosis and the incidence of subsequent symptomatic cancer. N Engl J Med 327: 1128-1133. https://doi.org/10.1056/NEJM199210153271604
[87]	Lee AYY, Levine MN (2003) Venous thrombosis and cancer: risks and outcomes. Circulation 107: I-17-I-21. https://doi.org/10.1161/01.CIR.0000078466.72504.AC
[88]	Dotsenko O, Kakkar AK (2006) Thrombosis and cancer. Ann Oncol 17: x81-x84. https://doi.org/10.1093/annonc/mdl242
[89]	Buller HR, Van Doormaal FF, Van Sluis GL, et al. (2007) Cancer and thrombosis: from molecular mechanisms to clinical presentations. J Thromb Haemost 5: 246-254. https://doi.org/10.1111/j.1538-7836.2007.02497.x
[90]	Sorensen HT, Mellemkjaer L, Olsen JH, et al. (2000) Prognosis of cancers associated with venous thromboembolism. N Engl J Med 343: 1846-1850. https://doi.org/10.1056/NEJM200012213432504
[91]	Tincani A, Taraborelli M, Cattaneo R (2010) Antiphospholipid antibodies and malignancies. Autoimmun Rev 9: 200-202. https://doi.org/10.1016/j.autrev.2009.04.001
[92]	Angchaisuksiri P (2016) Cancer-associated thrombosis in Asia. Thromb J 14: 26. https://doi.org/10.1186/s12959-016-0110-4
[93]	Zacharski LR, Henderson WG, Rickles FR, et al. (1979) Rationale and experimental design for the VA cooperative study of anticoagulation (warfarin) in the treatment of cancer. Cancer 44: 732-741. https://doi.org/10.1002/1097-0142(197908)44:2<732::AID-CNCR2820440246>3.0.CO;2-Y
[94]	Falanga A, Zacharski LR (2005) Deep vein thrombosis in cancer: the scale of the problem and approaches to management. Ann Oncol 16: 696-701. https://doi.org/10.1093/annonc/mdi165
[95]	Elyamany G, Alzahran AM, Burkhary E (2014) Clinical medicine insights: oncology. Cancer-associated thrombosis: an overview. Libertas Academia Limited 8: 129-137. https://doi.org/10.4137/CMO.S18991
[96]	Wallis CJD, Juvet T, Lee Y, et al. (2017) Association between use of antithrombotic medication and hematuria-related complications. JAMA 318: 1260-1271. https://doi.org/10.1001/jama.2017.13890
[97]	Petitjean A, Achatz MIW, Borresen-Dale AL, et al. (2007) TP53 mutations in human cancers: functional selection and impact on cancer prognosis and outcomes. Oncogene 26: 2157-2165. https://doi.org/10.1038/sj.onc.1210302
[98]	Surget S, Khoury MP, Bourdon JC (2013) Uncoverring the role of p53 splice variants in human malignancy: a clinical perspective. Onco Targets Ther 7: 57-68. https://doi.org/10.2147/OTT.S53876
[99]	Evans SC, Viswanathan M, Grier JD, et al. (2001) An alternative spliced HDM2 product increases p53 activity by inhibiting HDM2. Oncogene 20: 4041-4049. https://doi.org/10.1038/sj.onc.1204533
[100]	Wade M, Wong ET, Tang M, et al. (2006) Hdmx modulates the outcome of p53 activation in human tumor cells. J Biol Chem 281: 33036-33044. https://doi.org/10.1074/jbc.M605405200
[101]	Hamzehloie T, Mojarrad M, Hasanzadeh-Nazarabadi M, et al. (2012) Common human cancers and targeting the murine double minute 2-p53 Interaction for cancer therapy. Iran J Sci 37: 3-8.
[102]	White KA, Ruiz DG, Szpiech ZA, et al. (2017) Cancer-associated arginine-to- histidine mutations confer a gain in pH sensing to mutant proteins. Sci Signal 10: eaam9931. https://doi.org/10.1126/scisignal.aam9931
[103]	Zacharski LR, O'donnell JR, Henderson WG, et al. (1981) Effect of warfarin on survival in small cell carcinoma of the lung. Veterans administration study No. 75. JAMA 245: 831-835. https://doi.org/10.1001/jama.1981.03310330021017
[104]	Schulman S, Lindmarker P (2000) Incidence of cancer after prophylaxis with warfarin against recurrent venous thromboembolism. Duration of anticoagulation trial. N Engl J Med 342: 1953-1958. https://doi.org/10.1056/NEJM200006293422604
[105]	Tagalakis V, Tamim H, Blostein M, et al. (2007) Use of warfarin and risk of urogenital cancer: a population-based, nested case-control study. Lancet Oncol 8: 395-402. https://doi.org/10.1016/S1470-2045(07)70046-3
[106]	. Detection, evaluation and treatment of high blood cholesterol in adults (adult treatment panel III) final report (https://www.nhibi.nih.gov/files/docs/resources/heart/atp-3-cholesterol-full-report.pdf) (PDF). National Institutes of Health, National Heart, Lung, and Blood Institute. 1 September 2002. Retrieved 27 October 2008
[107]	Raso AG, Ene G, Miranda C, et al. (2014) Association between venous thrombosis and dyslipidemia. Med Clin (Barc) 143: 1-5.
[108]	Mi Y, Yan S, Lu Y, et al. (2016) Venous thromboembolism has the same risk factors as atherosclerosis. Medicine 96: 32(e4495). https://doi.org/10.1097/MD.0000000000004495
[109]	Stout JT, Caskey CT (1985) HPRT: gene structure, expression, and mutation. Ann Rev Genet 19: 127-148. https://doi.org/10.1146/annurev.ge.19.120185.001015
[110]	Patel PI, Framson PE, Caskey CT, et al. (1986) Fine structure of the human hypoxanthine phosphoribosyl transferase gene. Mol Cell Biol 6: 396-403. https://doi.org/10.1128/mcb.6.2.393-403.1986
[111]	Lesch M, Nyhan WL (1964) A familial disorder of uric acid metabolism and central nervous system function. Am J Med 36: 561-570. https://doi.org/10.1016/0002-9343(64)90104-4
[112]	Seegmiller JE, Rosenbloom FM, Kelley WN (1967) Enzyme defect associated with a sex-linked human neurological disorder and excessive purine synthesis. Science 155: 1682-1684. https://doi.org/10.1126/science.155.3770.1682
[113]	Morton NE, Lalouel JM (1977) Genetic epidemiology of Lesch-Nyhan disease. Am J Hum Genet 29: 304-311.
[114]	Fu R, Ceballos-Picot I, Torres RJ, et al. (2014) For the Lesch-Nyhan disease international study group. Genotype-phenotype correlation in neurogenetics: Lesch-Nyhan disease as a model disorder. Brain 137: 1282-1303. https://doi.org/10.1093/brain/awt202
[115]	Micheli V, Bertelli M, Jacomelli G, et al. (2018) Lesch-Nyhan disease: a rare disorder with many unresolved aspects. Medical University 1: 13-24. https://doi.org/10.2478/medu-2018-0002
[116]	Gottle M, Prudente CN, Fu R, et al. (2014) Loss of dopamine phenotype among midbrain neurone in Lesch-Nyhan disease. Ann Neurol 76: 95-107. https://doi.org/10.1002/ana.24191
[117]	Ceballos-Picot I, Mockel L, Potier MC, et al. (2009) Hypoxanthine-guanine phosphoribosyltransferase regulates early development programming of dopamine neurons: implication for Lesch-Nyhan disease pathogenesis. Hum Mol Genet 18: 2317-2327. https://doi.org/10.1093/hmg/ddp164
[118]	Connolly GP, Duley JA, Stacey NC (2001) Abnormal development of hypoxanthine-guanine phosphoribosyltransferase-deficient CNS neuroblastoma. Brain Res 918: 20-27. https://doi.org/10.1016/S0006-8993(01)02909-2
[119]	Stacey NC, Ma MHY, Duley JA, et al. (2000) Abnormalities in cellular adhesion of neuroblastoma and fibroblast models of Lesch-Nyhan syndrome. Neuroscience 98: 397-401. https://doi.org/10.1016/S0306-4522(00)00149-4
[120]	Harris JC, Lee RR, Jinnah HA, et al. (1998) Craniocerebral magnetic resonance imaging measurement and findings in Lesch-Nyhan syndrome. Arch Neurol 55: 547-553. https://doi.org/10.1001/archneur.55.4.547
[121]	Schretlen DJ, Varvaris M, Ho TE, et al. (2013) Regional brain volume abnormalities in Lesch-Nyhan disease and its variants: a cross-sectional study. Lancet Neurol 12: 1151-1158. https://doi.org/10.1016/S1474-4422(13)70238-2
[122]	Schretlen DJ, Varvaris M, Tracy D, et al. (2015) Brain white matter volume abnormalities in Lesch-Nyhan disease and its variants. Neurology 84: 190-196. https://doi.org/10.1212/WNL.0000000000001128
[123]	Kang TH, Friedmann T (2015) Alzheimer's disease shares gene expression aberrations with purinergic dysregulation of HPRT deficiency (Lesch-Nyhan disease). Neurosci Lett 590: 35-39. https://doi.org/10.1016/j.neulet.2015.01.042
[124]	Tuner PR, O'Connor K, Tate WP, et al. (2003) Roles of amyloid precursor protein and its fragments in regulating neural activity, plasticity and memory. Prog Neurobiol 70: 1-32. https://doi.org/10.1016/S0301-0082(03)00089-3
[125]	Zheng H, Koo EH (2006) The amyloid precursor protein: beyond amyloid. Mol Neurodegener 1: 5. https://doi.org/10.1186/1750-1326-1-5
[126]	Nguyen KV (2014) Epigenetic regulation in amyloid precursor protein and the Lesch-Nyhan syndrome. Biochem Biophys Res Commun 446: 1091-1095. https://doi.org/10.1016/j.bbrc.2014.03.062
[127]	Nguyen KV (2015) Epigenetic regulation in amyloid precursor protein with genomic rearrangements and the Lesch-Nyhan syndrome. Nucleosides Nucleotides Nucleic Acids 34: 674-690. https://doi.org/10.1080/15257770.2015.1071844
[128]	Nguyen KV, Nyhan WL (2017) Quantification of various APP-mRNA isoforms and epistasis in Lesch-Nyhan disease. Neurosci Lett 643: 52-58. https://doi.org/10.1016/j.neulet.2017.02.016
[129]	Imamura A, Yamanouchi H, Kurokawa T (1992) Elevated fibrinopeptide A (FPA) in patients with Lech-Nyhan syndrome. Brain Dev 14: 424-425. https://doi.org/10.1016/S0387-7604(12)80355-X
[130]	Riaz IB, Husnain M, Ateeli H (2014) Recurrent thrombosis in a patient with Lesch-Nyhan syndrome. Am J Med 127: e11-e12. https://doi.org/10.1016/j.amjmed.2014.01.033
[131]	Tewari N, Mathur VP, Sardana D, et al. (2017) Lesch-Nyhan syndrome: the saga of metabolic abnormality and self-injurious behavior. Intractable Rare Dis Res 6: 65-68. https://doi.org/10.5582/irdr.2016.01076
[132]	Canobbio I, Visconte C, Momi S, et al. (2017) Platelet amyloid precursor protein is a modulator of venous thromboembolism in mice. Blood 130: 527-536. https://doi.org/10.1182/blood-2017-01-764910
[133]	Townsend MH, Anderson MD, Weagel EG, et al. (2017) Non-small-cell lung cancer cell lines A549 and NCI-H460 express hypoxanthine guanine phosphoribosyltransferase on the plasma membrane. Onco Targets Ther 10: 1921-1932. https://doi.org/10.2147/OTT.S128416
[134]	Weagel EG, Townsend MH, Anderson MD, et al. (2017) Abstract 2149: Unusual expression of HPRT on the surface of the colorectal cancer cell lines HT29 and SW620. Cancer Res 77: 2149. https://doi.org/10.1158/1538-7445.AM2017-2149
[135]	Townsend MH, Felsted AM, Ence ZE, et al. (2017) Elevated expression of hypoxanthine guanine phosphoribosyltransferase within malignant tissue. Cancer Clin Oncol 6: 19. https://doi.org/10.5539/cco.v6n2p19
[136]	Townsend MH, Felsted AM, Burrup W, et al. (2018) Examination of hypoxanthine guanine phosphoribosyltransferase as a biomarker for colorectal cancer patients. Mol Cell Oncol 5: e1481810. https://doi.org/10.1080/23723556.2018.1481810
[137]	Townsend MH, Robison RA, O'Neill KL (2018) A review of HPRT and its emerging role in cancer. Med Oncol 35: 89. https://doi.org/10.1007/s12032-018-1144-1
[138]	Townsend MH, Shrestha G, Robison RA, et al. (2018) The expansion of targetable biomarkers for CAR T cell therapy. J Exp Clin Cancer Res 37: 163. https://doi.org/10.1186/s13046-018-0817-0
[139]	Townsend MH, Ence ZE, Felsted AM, et al. (2019) Potential new biomarkers for endometrial cancer. Cancer Cell Int 19: 19. https://doi.org/10.1186/s12935-019-0731-3
[140]	Pandey P, Sliker B, Peters HL, et al. (2016) Amyloid precursor protein and amyloid precursor-like protein 2 in cancer. Oncotarget 7: 19430-19444. https://doi.org/10.18632/oncotarget.7103
[141]	Nguyen KV (2019) β-Amyloid precursor protein (APP) and the human diseases. AIMS Neurosci 6: 273-281. https://doi.org/10.3934/Neuroscience.2019.4.273
[142]	Zheng H, Koo EH (2006) The amyloid precursor protein: beyond amyloid. Mol Neurodegener 1: 5. https://doi.org/10.1186/1750-1326-1-5
[143]	Nguyen KV (2015) The human β-amyloid precursor protein: biomolecular and epigenetic aspects. BioMol Concepts 6: 11-32. https://doi.org/10.1515/bmc-2014-0041
[144]	Xu F, Previti ML, Nieman MT, et al. (2009) AβPP/APLP2 family of Kunitz serine proteinase inhibitors regulate cerebral thrombosis. J Neurosci 29: 5666-5670. https://doi.org/10.1523/JNEUROSCI.0095-09.2009
[145]	Xu F, Davis J, Hoos M, et al. (2017) Mutation of the Kunitz-type proteinase inhibitor domain in the amyloid β-protein precursor abolishes its anti-thrombotic properties in vivo. Thromb Res 155: 58-64. https://doi.org/10.1016/j.thromres.2017.05.003
[146]	Di Luca M, Colciaghi F, Pastorino L, et al. (2000) Platelets as a peripheral district where to study pathogenetic mechanisms of Alzheimer disease: the case of amyloid precursor protein. Eur J Pharmacol 405: 277-283. https://doi.org/10.1016/S0014-2999(00)00559-8
[147]	Saonere JA (2011) Antisense therapy, a magic bullet for the treatment of various diseases: present and future prospects. J Med Genet Genom 3: 77-83.
[148]	Nguyen KV, Naviaux RK, Nyhan WL (2020) Lesch-Nyhan disease: I. Construction of expression vectors for hypoxanthine-guanine phosphoribosyltransferase (HGprt) enzyme and amyloid precursor protein (APP). Nucleosides Nucleotides Nucleic Acids 39: 905-922. https://doi.org/10.1080/15257770.2020.1714653
[149]	van Boven HH, Lane DA (1997) Antithrombin and its inherited deficiency states. Semin Hematol 34: 188-204.
[150]	Johnson EJ, Prentice CR, Parapia LA (1990) Premature arterial disease associated with familial antithrombin III deficiency. Thromb Haemost 63: 13-15. https://doi.org/10.1055/s-0038-1645677
[151]	Menache D, Grossman BJ, Jackson CM (1992) Antithrombin III: physiology, deficiency, and replacement therapy. Transfusion 32: 580-588. https://doi.org/10.1046/j.1537-2995.1992.32692367206.x
[152]	Cordell HJ (2002) Epistasis: what it means, what it doesn't mean, and statistical method to detect it in humans. Hum Mol Genet 11: 2463-2468. https://doi.org/10.1093/hmg/11.20.2463
[153]	Moore JH (2003) The ubiquitous nature of epistasis in determining susceptibility to common human diseases. Hum Hered 56: 73-82. https://doi.org/10.1159/000073735
[154]	Riordan JD, Nadeau JH (2017) From peas to disease: modifier genes, network resilience, and the genetics of health. Am J Hum Genet 101: 177-191. https://doi.org/10.1016/j.ajhg.2017.06.004
[155]	Coutard B, Valle C, de Lamballerie X, et al. (2020) The spike glycoprotein of the new coronavirus 2019-nCoV contains a furin-like cleavage site in CoV of the same clade. Antiviral Res 176: 104742. https://doi.org/10.1016/j.antiviral.2020.104742
[156]	Nguyen KV (2021) Problems associated with antiviral drugs and vaccines development for COVID-19: approach to intervention using expression vectors via GPI anchor. Nucleosides Nucleotide Nucleic Acids 40: 665-706. https://doi.org/10.1080/15257770.2021.1914851

This article has been cited by:

玉芳王, Multiplicity of Positive Solutions forNonlinear Elliptic Problems inExterior Domain, 2024, 14, 2160-7583, 605, 10.12677/PM.2024.145214

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)