Yannis-Nicolas François

Michael Biacchi

Nassur Said

Charly Renard

Alain Beck

Rabah Gahoual

Emmanuelle Leize-Wagner

Characterization of Cetuximab F c/2 aggregates by Offline CZE-MS

Keywords: Capillary Electrophoresis-Mass spectrometry, Monoclonal antibody, Aggregates, nanoElectrospray

Monoclonal antibody (mAb) therapeutics attract the largest concern due to their strong therapeutic potency and specificity. The Fc region of mAbs is common to many new biotherapeutics as biosimilar, antibody drug conjugate or fusion protein. Fc region has consequences for Fc-mediated effector functions that might be desirable for therapeutic applications. As a consequence, there is a continuous need for improvement of analytical methods to enable fast and accurate characterization of biotherapeutics. Capillary zone electrophoresis-Mass spectrometry couplings (CZE-MS) appear really attractive methods for the characterization of biological samples. In this report, we used CZE-MS systems developed in house and native MS infusion to allow precise middle-up characterization of Fc/2 variant of cetuximab. Molecular weights were measured for three Fc/2 charge variants detected in the CZE separation of cetuximab subunits. Two Fc/2 C-terminal lysine variants were identified and separated. As the aim is to understand the presence of three peaks in the CZE separation for two Fc/2 subunits, we developed a strategy using CZE-UV/MALDI-MS and CZE-UV/ESI-MS to evaluate the role of N-glycosylation and C-terminal lysine truncation on the CZE separation. The chemical structure of Nglycosylation expressed on the Fc region of cetuximab does not influence CZE separation while Cterminal lysine is significantly influencing separation. In addition, native MS infusion demonstrated the characterization of Fc/2 dimers at pH 5.7 and 6.8 and the first separation of these aggregates using CZE-MS.

INTRODUCTION

Although monoclonal antibodies (mAbs) were introduced as treatments against disease in the late 1980, they currently represent the most rapidly growing category of therapeutic molecule [START_REF] Beck | Strategies and challenges for the next generation of therapeutic antibodies[END_REF][START_REF] Reichert | Marketed therapeutic antibodies compendium[END_REF][START_REF] Beck | Analytical characterization of biosimilar antibodies and Fc-fusion proteins[END_REF]. MAbs are highly complex glycoproteins potentially displaying many naturally-occurring molecular microheterogeneities [START_REF] Arnold | The impact of glycosylation on the biological function and structure of human immunoglobulins[END_REF]5]. The Fc region of mAbs is common to many new platforms as biosimilar, antibody drug conjugate or fusion protein. This region of the protein is conditioning the antibody isotype and particularly is responsible for effector functions such as antibody-dependent cytotoxicity (ADCC), complement-dependent cytotoxicity (CDC) like phagocytosis [START_REF] Kinet | Fc Receptors[END_REF]. Additionnally it facilitates purification and can also contribute to improved solubility and stability [START_REF] Torres | The immunoglobulin constant region contributes to affinity and specificity[END_REF]. The presence of an Fc region has consequences for Fc-mediated effector functions that might be desirable for therapeutic applications.

For successful delivery of biotherapeutic to patient, one of the major issue is aggregates formation.

Aggregates in formulated therapeutic products can cause unwanted immune responses and loss of biological activity [START_REF] Braun | Protein Aggregates Seem to Play a Key Role Among the Parameters Influencing the Antigenicity of Interferon Alpha (IFN-α) in Normal and Transgenic Mice[END_REF][START_REF] Schellekens | Bioequivalence and the immunogenicity of biopharmaceuticals[END_REF]. Literature suggests that, depending on the nature of mAbs, mAbs aggregation in human might be mediated primarily by unfolding of CH2 domain on the Fc region [START_REF] Wu | Competing aggregation pathways for monoclonal antibodies[END_REF][START_REF] Andersen | Aggregation of a multidomain protein: A coagulation mechanism governs aggregation of a model IgG1 antibody under weak thermal stress[END_REF] and Fab domain [START_REF] Wu | Competing aggregation pathways for monoclonal antibodies[END_REF][START_REF] Brummitt | Nonnative Aggregation of an IgG1 Antibody in Acidic Conditions: Part 1. Unfolding, Colloidal Interactions, and Formation of High-Molecular-Weight Aggregates[END_REF]. As a consequence, there is a continuous need for improvement of analytical methods to enable fast and accurate characterization. Mass spectrometry (MS) generally coupled with separation methods plays a key role in the characterization of therapeutic mAbs [START_REF] Beck | Biosimilar, Biobetter, and Next Generation Antibody Characterization by Mass Spectrometry[END_REF] . Depending on the protein structure level considered, characterization is performed following different strategies as intact molecular weight (MW) measurement, top-down, middle-up and bottom-up approaches [START_REF] Beck | Biosimilar, Biobetter, and Next Generation Antibody Characterization by Mass Spectrometry[END_REF][START_REF] Beck | Characterization of Therapeutic Antibodies and Related Products[END_REF][START_REF] Beck | Cutting-edge mass spectrometry characterization of originator, biosimilar and biobetter antibodies[END_REF][START_REF] Gahoual | Monoclonal antibodies biosimilarity assessment using transient isotachophoresis capillary zone electrophoresis-tandem mass spectrometry[END_REF][START_REF] Gahoual | Full Antibody Primary Structure and Microvariant Characterization in a Single Injection Using Transient Isotachophoresis and Sheathless Capillary Electrophoresis-Tandem Mass Spectrometry[END_REF]. Capillary zone electrophoresis (CZE) has been demonstrated to be an useful and powerful separation method for the characterization of intact proteins [START_REF] Han | Sheathless Capillary Electrophoresis-Tandem Mass Spectrometry for Top-Down Characterization of Pyrococcus furiosus Proteins on a Proteome Scale[END_REF][START_REF] Sun | Fast Top-Down Intact Protein Characterization with Capillary Zone Electrophoresis-Electrospray Ionization Tandem Mass Spectrometry[END_REF][START_REF] Zhao | Coupling Capillary Zone Electrophoresis with Electron Transfer Dissociation and Activated Ion Electron Transfer Dissociation for Top-Down Proteomics[END_REF]. In a previous work, we developed the first analysis of intact mAb charge variant by CZE using a matrix-assisted laser desorption/ionization-MS (MALDI) detection [START_REF] Biacchi | Analysis of monoclonal antibody by a novel CE-UV/MALDI-MS interface[END_REF]. In 2015, Redman et al published a remarkable paper on characterization of intact mAb variants using microfluidic CZE-ESI devices [START_REF] Redman | Integrated Microfluidic Capillary Electrophoresis-Electrospray Ionization Devices with Online MS Detection for the Separation and Characterization of Intact Monoclonal Antibody Variants[END_REF]. However, every studies described mAbs characterization without any aggregation phenomenon occurring. Recently, we developed a method for the middle-up characterization of cetuximab glycoforms using off-line CZE-UV/ESI-MS [START_REF] Biacchi | Glycoform Separation and Characterization of Cetuximab Variants by Middle-up Off-Line Capillary Zone Electrophoresis-UV/Electrospray Ionization-MS[END_REF].

In this report, we used CZE-MS systems developed in house and native MS infusion to allow precise middle-up characterization of Fc/2 variant. The mAb selected was cetuximab which is human/murine chimeric IgG-1 directed against the epidermal growth factor receptor (EGFR) overexpressed in advanced-stage EGFR positive colorectal cancer. Cetuximab was approved in the US and EU in 2004 and 2005. Cetuximab contains two glycosylation sites both on the heavy chain (HC) [START_REF] Beck | Biosimilar, Biobetter, and Next Generation Antibody Characterization by Mass Spectrometry[END_REF]. Moreover, it has a large number of micro-heterogeneities such as PTMs including methionine oxidation, asparagine deamidation or isomerization of aspartic acid. This mAb also has one C-terminal lysine truncation. These cumulated features make cetuximab an ideal sample for the characterization of mAbs variants by a middle-up approach. In order to cleave the protein in relatively large fragments and distribute Nglycosylation sites over different fragments generated, IdeS proteolysis was employed. IdeS is a cysteine endopeptidase enzyme naturally secreted by Streptococcus pyogenes [START_REF] Wenig | Structure of the streptococcal endopeptidase IdeS, a cysteine proteinase with strict specificity for IgG[END_REF], IdeS enzymatic reaction cleaves cetuximab between the two consecutive glycine residues present in the hinge region to obtain two types of fragments (Fc/2 and F(ab')2) each one carrying a N-glycosylation site. IdeS has demonstrated an exceptional specificity in IgG proteolysis on contrary to papain for instance, therefore it has been consequently used to ease and improve mAbs characterization [START_REF] Vincents | Enzymatic characterization of the streptococcal endopeptidase, IdeS, reveals that it is a cysteine protease with strict specificity for IgG cleavage due to exosite binding[END_REF]. IdeS characteristics regarding specificity alongside to proteolysis kinetics are explaining the success of this enzyme in mAbs characterization [START_REF] Ayoub | Correct primary structure assessment and extensive glyco-profiling of cetuximab by a combination of intact, middle-up, middle-down and bottom-up ESI and MALDI mass spectrometry techniques[END_REF]. We developed a strategy using CZE-UV/MALDI-MS and CZE-UV/ESI-MS to evaluate the role of N-glycosylation and C-terminal lysine truncation on the CZE separation. In addition, this approach demonstrated the characterization of Fc/2 dimers at pH 5.7 and 6.8 and the first separation of these aggregates using CZE-UV off-line hyphenated to ESI-MS after Fraction collection. A better understanding of these phenomenon could improve the characterization of Fc region and their potential non-covalent interactions [START_REF] Wu | Weak protein interactions and pH-and temperature-dependent aggregation of human Fc1[END_REF].

Materials and methods

2.1. Chemicals. Methanol (HPLC gradient grade) and acetic acid (100%) were obtained from VWR (Radnor, PA, USA). Ammonium acetate (>98%), sodium hydroxide, ε-Amino-caproic acid (>98%), hydroyxypropylcellulose Mw 100000 (HPC), carboxypeptidase B (CP-B) and formic acid (>98%) were purchased from Sigma-Aldrich (Saint Louis, MO, USA). Water used to prepare buffers and sample solutions was obtained using an ELGA purelab UHQ PS water purification system (Bucks, UK). IdeS (Immunoglobulin-degrading enzyme of Streptococcus pyogenes) also named FabRICATOR and IgGZERO were purchased from Genovis (Lund, Sweden). Cetuximab (Erbitux®, Merck KGaA, Darmstadt, Germany) is a sterile, preservative-free solution for intravenous infusion containing 5 mg/mL of cetuximab. Additionally to the mAbs it contains sodium chloride, glycine, polysorbate 80, citric acid monohydrate, sodium hydroxide and water for injections.

Middle-up sample preparation.

Cetuximab was cleaved in the hinge region using limited proteolysis by IdeS (FabriCATOR, Genovis) to obtain two Fc/2 fragments (theoretical pI 7.74) and one F(ab')2 fragment (theoretical pI 7.78). Sample was diluted using 147.25µL of 50 mM sodium phosphate, 150 mM NaCl, pH 6.60, to a final concentration of 1 µg/µL. A volume of 2.25 µL of IdeS (67 units/µL) was added to the sample which was left at 37°C for 30 min. After digestion completion, sample was desalted using Amicon centrifugal filters (cut off = 10,000 Da) in pure water at 10°C and 14,000 g for 20 min. After desalting step, sample volume recovered was about 10 µL. Sample was finally diluted to a final concentration of 5 µg/µL in a total volume of 30 µL of pure water.

Carboxypeptidase B (CP-B) treatment.

Cetuximab was diluted using 145.75 µL with 10 mM phosphate buffer, 150 mM NaCl, pH 7.4, to a concentration of 1 µg/L. Volumes of 2 µL of CP-B (1mg/mL) and 2.25 µL of IdeS (67 units/µL) were added to the sample which was left at 37°C for 30 min. After digestion completion, sample was treated as described in the middle-up sample preparation.

2.4. Deglycosylation protocol. Cetuximab was diluted using 141.75 µL with 10 mM phosphate buffer, 150 mM NaCL, pH 7.4, to a concentration of 1 µg/µL. Volumes of 6 µL of IgGZERO (20 units/µL) and 2.25 µL of IdeS (67 units/µL) were added to the sample which was left at 37°C for 30 min. After digestion completion, sample was treated as described in the middle-up sample preparation.

Capillary electrophoresis.

The CZE experiments were carried out on a P/ACE MDQ TM CE system from Sciex Separation (Brea, CA) equipped with a UV detection, a temperature controlled autosampler and a power supply able to deliver up to 30 kV. A 32 Karat TM 8.0 (Sciex Separation, Brea, CA) was used for instrument control, data acquisition and data handling. Polymicro bare fused-silica capillaries of 75 µm i.d., 375 o.d. (75.5 cm effective length, 82 cm total length) were obtained from Photonlines (St-Germain-en-Laye, France). New capillaries were conditioned by successive flushes with 1M and 0.1M NaOH and then with water under a pressure of 30 psi for 10 min each. The temperature in the capillary cartridge and autosampler was set to 25 °C. The acquisition rate was 10 points/s. Capillaries were rinsed with water and dried by air when not in use. UV absorbance remained fixed at 200 nm. Voltage applied at 20 kV with a ramp duration of 0.17 min and injection sample conditions were 0.5 psi for 50s.

Concerning modified capillaries, capillaries were coated in house with hydroyxypropylcellulose Mw 100000 (HPC) following the protocol described by Shen et al [START_REF] Shen | High-resolution capillary isoelectric focusing of proteins using highly hydrophilic-substituted cellulose-coated capillaries[END_REF]. 5% HPC in pure water (w/v) was prepared to apply capillary coating. Coating durability is around 20 runs without recoating step. For cetuximab separation, inlet BGE 200 mM ε-Amino-caproic acid (EACA)-ammonium acetate 25 mM pH 5.70 and outlet BGE ammonium acetate 25 mM pH 5.70 have been used as separation conditions.

Injection volumes have been calculated using CEToolbox application (Pansanel, GooglePlay).

CZE/Fraction Collection

Interface. This interface have been described in a previous study [START_REF] Biacchi | Analysis of monoclonal antibody by a novel CE-UV/MALDI-MS interface[END_REF].

Briefly, automated off-line coupling of CZE to MS was set up by using a homemade modified automatic spotting device Proteineer FC (Bruker Daltonics, Bremen, Germany) for the sheath flow-assisted spotting from the CZE capillary outlet onto a fraction collection target. The original set up of the UV cell in the P/ACE MDQ TM (Sciex Separation, Brea, CA) was delocalized outside the CE instrument in order to allow simultaneous UV detection and fractions collection. Hystar 3.2 (Bruker Daltonics, Bremen, Germany) was used for Proteineer FC control.

CESI-MS as nanoESI infusion platform.

The infusion experiments were carried out with a PA 800 plus capillary electrophoresis (CE) system from Sciex Separation (Brea, CA). Prototype fused-silica capillaries (total length 95 cm; 30 µm i.d.) whose outlet end (about 3 cm) was etched with hydrofluoric acid were provided by Sciex Separation (Brea, CA, USA). New capillaries were initially conditioned by flushing them for 10 min with MeOH, 10 min with 0.10 M sodium hydroxide, 10 min with 0.10 M hydrochloric acid and finally with water for 20 min each flushing step being conducted at 75 psi (5.17 bar).

MALDI-TOF-MS.

The matrix was prepared by dissolving 2,5-dihydroxybenzoic acid (DHB) (2g/L) in 0.1% trifluoroacetic acid/acetonitrile (TFA/ACN) (30/70)(v/v). Fraction collection was realized using Ground Steel MALDI target (Bruker Daltonics, Bremen, Germany). Mass spectra of the CZE fractions were recorded using an Autoflex II MALDI-TOF (Bruker Daltonics, Bremen, Germany), operating in reflector mode and with FlexControl software. Positively charged ions were detected and sums of 1500 single-shot spectra were acquired automatically from each sample by using the AutoXecute software.

Data processing was performed with FlexAnalysis 3.0 (Bruker Daltonics, Bremen, Germany). All spectra were calibrated by external calibration using Protein calibration standard I (Bruker Daltonics, Bremen, Germany) for intact protein separation.

ESI-TOF-MS.

For nanoESI-MS infusion experiments the PA800 plus system was coupled to a maxis 4G (Bruker Daltonics, Bremen, Germany). MS transfer parameters were optimized using the actual sample directly infused via the CE system using a pressure of 5 psi (340 mbar). MS parameters were optimized so that high m/z ions could be properly transferred to the TOF mass analyzer while avoiding fragmentation. Ion funnels RF were set at values of 300 Vpp and 400 Vpp. The electrospray voltage (capillary voltage) was typically ranging from -1.2 to -1.8 kV. Dry gas was set at 1.5 L/min and source temperature at 180°C. Data processing was performed with DataAnalysis 4.0 (Bruker Daltonics, Bremen, Germany). Deconvolution of the mass spectra was performed based on maximum entropy analysis using ESI Compass 1.3 Maximum Entropy Deconvolution Option in DataAnalysis 4.0. All spectra were calibrated by external calibration using Pepmix (Bruker Daltonics, Bremen, Germany) and cesium iodide (CsI) from Sigma-Aldrich (Saint-Louis, MO, USA).

Results and discussions

Cetuximab is a chimeric mouse-human IgG1 known to bear 2 N-glycosylation sites on each HC [START_REF] Ayoub | Correct primary structure assessment and extensive glyco-profiling of cetuximab by a combination of intact, middle-up, middle-down and bottom-up ESI and MALDI mass spectrometry techniques[END_REF]. As every mAbs, cetuximab can exhibit a wide variety of micro-heterogeneities particularly due to PTMs. Furthermore, this mAb can also experience HC C-terminal lysine truncation. In a preliminary experiment, direct nanoESI-MS infusion of cetuximab IdeS digest was performed. Results proved equivalent signal abundances for Fc/2 C-terminal lysine variants which involves two distinct profiles for the Fc/2 domains (Fig. S-1). With this information, we expected to detect at least two peaks corresponding to Fc/2 fragments due to a loss of a +1 charge associated with C-terminal lysine truncation.

Cetuximab IdeS digest was separated and characterized by offline CZE-UV/ESI-MS using an instrumental setting developed in house, described in a previous work [START_REF] Biacchi | Glycoform Separation and Characterization of Cetuximab Variants by Middle-up Off-Line Capillary Zone Electrophoresis-UV/Electrospray Ionization-MS[END_REF]. Briefly, assymetric CZE conditions was based on inlet background electrolyte (BGE) composed of ε-amino-caproic acid (EACA) 200 mM/ammonium acetate 25 mM, pH 5.7 and outlet BGE composed of ammonium acetate 25 mM, pH 5.7. Hydroxypropylcellulose (HPC) coating was used to minimize analyte adsorption on the capillary wall and reduce the electroosmotic flow (EOF). After deposition, each fraction was collected and evaporated. Before infusion, dry samples were reconstituted in 2μL of ACN/H20/FA (50/49/1, v/v) in order to enhance sample ionization. Thus, each fraction was infused into the MS using a flow rate of 10 nL/min in order to enhance sensitivity.

As emphasized in Fig. 1-b, CZE separation obtained shows three consecutive peaks completely separated corresponding to the Fc/2 region. Deconvoluted mass spectra of each fraction generated multiple masses corresponding to the expected presence of different neutral glycosylation on Fc/2 fragments (Fig. 1-b). In addition, the comparison with the direct infusion experiment (Fig. S-1) demonstrates that the two Fc/2 fragments corresponding to intact and with one lysine truncation (Fc/2 and Fc/2-K) are detected and separated. Deconvoluted mass spectra corresponding to peak 1 and peak 3 show a difference of 128 Da between mass peak 25359.1 Da corresponding to the most abundant Fc/2 glycoform (G0F) and mass peak 25231.1 Da corresponding to G0F for Fc/2-K. Thus, we demonstrate the complete separation of these two fragments (Fig. 1-b). Regarding deconvoluted mass spectra of peak 2 (Fig. 1-b 2), it shows a nearly equal abundance mixture of Fc/2 and Fc/2-K variants. Deconvoluted mass spectra corresponding to peak 2 lead to the same MWs as peak 1 and peak 3 and could be attributed to either an overlap during the fraction collection process or by cumulative PTMs which could lead to a variation of conformation without mass shifts. Moreover, these phenomenon could also be explained by a possible aggregation of Fc/2 fragments.

Validation of the CZE-UV/ESI-MS fraction collection process.

The fraction collection process developed in this study based on the apparent mobility of each peak, allows theoretically to recover the total quantity of each variant present in the peak. However the offline coupling strategy developed do not permit to represent the separation using MS spectra and then to observe possible diffusion. Regarding raw MS spectra (Fig. S-2) and based on the hypothesis of an equivalent ionization efficiency of all Fc/2 variants, potential carryover effect due to peak broadening or excessive diffusion should be eliminated. However, in the aim to confirm the absence of overlap during the fraction collection, experiments using CZE-UV/MALDI-MS coupling have been performed. Same experimental conditions described previously have been implemented. Only deposition process has been changed to allow a fraction collection directly on a MALDI target plate with a deposition time interval of 1 minute between each fraction. Fraction collection was performed between 43 and 55 minutes according to the electropherogram represented in Fig. 2-a. Fig. 2-b emphasizes the reconstruction of the separation using MS spectra of each fraction. First of all, the CZE/MALDI-MS reconstruction exhibits three distinct peaks corresponding to the CZE separation. MS spectra of each peak in CZE show a value of 25 kDa corresponding to the expected Fc/2 variants. However due to presence of other sources of micro-heterogeneities and the limited resolution of the MALDI-MS instrument used, it was impossible to deduce the average mass of the charge variants confidently. In the other hand, the CZE/MALDI-MS profile (Fig. 2-b) confirms the good agreement between the UV detection time and the deposition time. Based on the difference of protein apparent mobilities described by the equation:

µ 𝑎𝑝𝑝 = L l t 𝑚 V
with µapp the apparent mobility which is the sum of effective mobility and residual electroosmotic flow, L and l the total capillary length and length to detection window, respectively, V the applied voltage and tm the migration time. Deposition time td is then calculated by the equation:

𝑡 𝑑 = 𝐿 2 µ 𝑎𝑝𝑝 𝑉
The localization of each Fc/2 variants shows a time shift between the UV detections and the spotting process due to UV localization at 6.5 cm from the capillary outlet. These results confirm without any remaining ambiguity the absence of carryover effect and diffusion phenomenon during the fraction collection process, demonstrating the accuracy and robustness of the instrumental setting developed.

As a consequence, characteristics of the analytical strategy developed to characterize Fc/2 fragments cannot be doubted. Therefore it is necessary to focus on cetuximab structure and potential modifications on the fragments generated during IdeS proteolysis in order to conclude on the CZE separation described previously (Fig. 1-a).

Influence of N-glycosylation on Fc/2 variants separation

An important class of modification that can significantly impact the immunogenic properties of cetuximab in terms of pharmacokinetic and pharmacodynamic is glycosylation [START_REF] Ayoub | Correct primary structure assessment and extensive glyco-profiling of cetuximab by a combination of intact, middle-up, middle-down and bottom-up ESI and MALDI mass spectrometry techniques[END_REF]. Cetuximab contains two sites of glycosylation on the HC: one is located in the Fc/2 domain (Asn 299 ) and the second is located in the F(ab')2 domain on Asn 88 [START_REF] Beck | Biosimilar, Biobetter, and Next Generation Antibody Characterization by Mass Spectrometry[END_REF]. Cetuximab glycosylation profile has been extensively described in particular as it is expressing glycoforms responsible for immunogenic reaction [START_REF] Ayoub | Correct primary structure assessment and extensive glyco-profiling of cetuximab by a combination of intact, middle-up, middle-down and bottom-up ESI and MALDI mass spectrometry techniques[END_REF].

Regarding Fc/2 domain, branched glycan structures are primarily composed of fucose, Nacetylglucosamine, and mannose but can differ in the number of terminal galactose, mannose or Nacetylglucosamine residues. Glycan structure with the addition of galactose moieties involves a mass increase of 162 Da but does not induce a change in net charge [START_REF] Liu | Heterogeneity of monoclonal antibodies[END_REF][START_REF] Raju | Glycosylation in the Fc domain of IgG increases resistance to proteolytic cleavage by papain[END_REF]. In a previous work we demonstrated that concerning glycoform with a difference of one galactose residue, middle-up approach does not allow to observe baseline separations [START_REF] Biacchi | Glycoform Separation and Characterization of Cetuximab Variants by Middle-up Off-Line Capillary Zone Electrophoresis-UV/Electrospray Ionization-MS[END_REF]. The mass of galactose moiety accounts for only ≈0.6% of the Fc/2 domain and ≈0.2% of the F(ab')2 domain. Thus mass difference should be greater or involve a significant conformation change to enable a significant effective mobility shift. To confirm these theoretical considerations, analysis of deglycosylated sample of cetuximab by CZE-UV/ESI-MS was performed in order to evaluate the role of glycans in the separation process. Before IdeS digestion of cetuximab, a deglycosylation step using IgGZERO has been added to the digestion protocol. The aim is to remove all glycans in the Asn 299 . IgGZERO is an endoglycosidase specific for Nglycans on the Fc domain of IgGs which are cleaved after the first Glc-NAc. Following IgGZERO and IdeS treatment, the most intense ion series is corresponding to expected cetuximab Fc/2 and Fc/2-K subunits with GlcNac-Fuc remaining (Fig. 4-b andTable S-1). CZE separation, represented by Fig. 3-B, contains the three peaks corresponding to Fc/2 subunits with the same profile in terms of migration times and resolution compared to the original separation of the sample non treated with IgGZERO enzyme (Fig. 3-A). With these results, we can conclude that glycan nature expressed on Asn 299 residue does not impact significantly the CZE separation of Fc/2 variants. Effect of others microheterogeneities like asparagine deamidation or methionine oxidation are quite difficult to observe on the intact protein or using limited proteolytic digestion because of the number of these modifications, their localization and the difficulty to control the level of modification in the proteins. Indeed, some parameters like storage conditions or digestion pH can induce endogenous asparagine deamidation or methionine oxidation.

Influence of C-terminal lysine truncation on Fc/2 variants separation

The presence of the lysine in HC C-terminal position is essential to understand the separation of Fc/2 variants. Indeed, in this work, we demonstrated the separation of the two Fc/2 fragments corresponding to intact and with one lysine truncation (Fc/2 and Fc/2-K). Peak 1 and peak 3 (Fig. 3-A) correspond to the most abundant Fc/2 glycoforms and Fc/2-K glycoforms respectively. However for peak 2, a nearly equal abundance mixture of Fc/2 and Fc/2-K have been observed. To evaluate the role of lysine truncation in the separation process, IdeS digestion protocol has been modified to perform a total truncation of C-1). Concerning CZE conditions, charge variant deconvoluted mass spectrum generated multiple masses between 50.4 kDa and 51.2 kDa (Fig. 5-b). These peaks correspond to the formation of protein-protein aggregates due to noncovalent interactions between Fc/2 subunits. Indeed, mass peak 50463.4 Da corresponds to the mass of Fc/2-K subunits homodimer, mass peak 50591.8 Da corresponds to the heterodimer between Fc/2 and Fc/2-K subunits and mass peak 50721.2 Da corresponds to Fc/2 subunit homodimer (Table S-2). Moreover, other mass peaks between 50.4 kDa and 51.2 kDa correspond to the expected presence of the combination of neutral glycosylation variants of these different dimers (Table S-2). In addition, no deconvoluted mass spectrum have been observed at 25 kDa that excludes the presence of free Fc/2 variants in ammonium acetate 25mM at pH 5.7. These results allow the explanation of the presence of three different peaks corresponding to Fc/2 variants in CZE separation. Indeed, peak 1 (Fig. 1-a) corresponds to Fc/2 homodimer, peak 2 corresponds to Fc/2/Fc/2-K heterodimer and peak 3 correspond to Fc/2-K homodimer. To our knowledge, this is the first time that aggregates of cetuximab Fc/2 subunits have been separated and characterized. Finally, a last study consisted to infuse the same sample in condition classically described in the literature as native conditions [START_REF] Gahoual | Novel sheathless CE-MS interface as an original and powerful infusion platform for nanoESI study: from intact proteins to high molecular mass noncovalent complexes[END_REF][START_REF] Nettleton | Protein subunit interactions and structural integrity of amyloidogenic transthyretins: Evidence from electrospray mass spectrometry[END_REF][START_REF] Benesch | Separating and visualisiing protein assemblies by means of preparative mass spectrometry and microscopy[END_REF][START_REF] Touboul | How to Deal with Weak Interactions in Noncovalent Complexes Analyzed by Electrospray Mass Spectrometry: Cyclopeptidic Inhibitors of the Nuclear Receptor Coactivator 1-STAT6[END_REF]. To be in agreement with CZE condition, 25 mM ammonium acetate at pH 6.8 was selected. Obtained MS profiles were exactly the same as those observed using CZE conditions. This confirms the presence of

Conclusions

To summarize, we report here the use of off-line CZE-UV/ESI-MS for the middle-up characterization of

Fc/ 2

 2 dimers in native conditions and exclude a possible major role of EACA in the protein-protein aggregation process. Results obtained from CZE-UV/MALDI-MS and CZE-UV/ESI-MS experiments clearly demonstrate the formation of Fc/2 fragments following IdeS proteolytic digestion of mAbs. This observation is particularly important from an analytical chemistry point of view. As the outstanding selectivity of IdeS has favored its wide use among the scientific community and the formation of aggregates, even if reversible, it should be taken into account in the development of innovative analytical methods.

Fc/ 2

 2 variant of cetuximab. To validate the method, developed CZE offline instrumental setting was implemented to perform direct deposition of collected fractions on a MALDI target and used to characterize IdeS proteolytic digest fragments by MALDI-TOF MS. Results obtained demonstrated successful CZE-UV/MALDI-MS characterization. MS spectra were in complete agreement with the separation observed by the intermediate of the UV detection. Additionally data generated allowed to exclude overlapping during the collection and deposition process validating the instrumental approach regarding accuracy and robustness. In order to investigate further the separation of three different species corresponding to Fc/2 variants, CZE-UV/ESI-MS experiments were performed on cetuximab treated consequently with igGZERO and IdeS enzymes to permit the removal of N-glycosylation present in the Fc region of the protein. Results showed that chemical structure of N-glycosylation expressed on the Fc region of cetuximab does not influence CZE separation as both revealed similar UV electropherograms. A similar analysis was then realized on cetuximab treated successively with CP-B and IdeS in order to systematically remove C-terminal lysine from the Fc/2 fragments generated. Results from the CZE separation exhibited the absence of two peaks compared to the same separation using cetuximab solely treated with IdeS. As expected from the CZE conditions employed, C-terminal lysine is significantly influencing separation. Finally direct nanoESI-MS infusion of the collected fractions from CZE separation were performed using different buffering systems, one which is similar to CZE background electrolyte composed of ammonium acetate 25 mM, pH 5.70, and in parallel in denaturing conditions favoring ionization efficiency and signal sensitivity. In similar conditions as CZE separation, MS allowed to identify without ambiguity the nature of each analyte separated by CZE. MS spectra obtained revealed the first specie to be Fc/2 homodimer while the second specie detected was Fc/2/Fc/2-K heterodimer and the last one being Fc/2-K homodimer. Results demonstrated to be in agreement regarding the influence of C-terminal lysine on the separation but additionally showed the formation of Fc/2 aggregates as a consequence of IdeS proteolytic digestion. Results from the different experiments demonstrated the relevance of this CZE-UV/ESI-MS analytical method developed to perform the comprehensive middle-up characterization of mAbs. Indeed it was possible to conclude of the formation of Fc/2 aggregates generated consequently from cetuximab IdeS treatment. As this enzyme recently tends to be widely used, due its exceptional specificity, for mAbs characterization, aggregation related to sample treatment appears as a new insight which should be taken into account during method development. The implementation of an electrophoretic separation prior to the MS analysis showed its interests in this work as separation efficiency provided by CZE allowed to completely separate the different types of aggregates while showing the possibility to maintain these aggregates. Moreover, the Fc region of mAbs is common to many new platforms and these results could improve the characterization of Fc region of biotherapeutics and their potential non-covalent interactions.

Fig. 1 .

 1 Fig. 1. (a) Off line CZE-UV/ESI-MS separation of middle-up cetuximab charge variants. (b) Deconvoluted mass spectra for each Fc/2 variant. The MS peak were labeled with the correspondent glycoform. Experimental conditions: Inlet BGE: EACA 200 mM ammonium acetate 25 mM pH 5.70 and outlet BGE: ammonium acetate 25 mM pH 5.70; HPC-coated capillary, total/effective length 82/75.5 cm x 75 µm i.d.; voltage, 20 kV; UV absorbance at 200 nm; MS condition: See Experimental section; sample, IdeS digest of cetuximab (5 µg/µL); sample injection 0.5 psi 50s.

Fig. 2 .Fig. 3 .

 23 Fig. 2. CZE-UV off-line coupled to MALDI-MS via fractionation for a middle-up characterization of Cetuximab Fc/2 variants. (a) CZE-UV electropherogram. (b) Analysis of CZE-UV fractions by MALDI-MS. CE Experimental conditions: See Fig. 1; MS condition: See Experimental section; sample, IdeS digest of cetuximab (5 µg/µL)

Fig. 4 .

 4 Fig. 4. Fc/2 variants deconvoluted mass spectra of (a) IdeS digest cetuximab, (b) IgGZERO+IdeS digest cetuximab and (c) CP-B+IdeS digest cetuximab. Experimental conditions: 50%/49%/1%, acetonitrile/water/formic acid (v/v/v); sample concentration 1 µM.

Caption

Fig. 1 .

 1 Fig. 1. (a) Off line CZE-UV/ESI-MS separation of middle-up cetuximab charge variants. (b) Deconvoluted mass spectra for each Fc/2 variant. The MS peak were labeled with the correspondent glycoform. Experimental conditions: Inlet BGE: EACA 200 mM ammonium acetate 25 mM pH 5.70 and outlet BGE: ammonium acetate 25 mM pH 5.70; HPC-coated capillary, total/effective length 82/75.5 cm x 75 µm i.d.; voltage, 20 kV; UV absorbance at 200 nm; MS condition: See Experimental section; sample, IdeS digest of cetuximab (5 µg/µL); sample injection 0.5 psi 50s.

Fig. 2 .Fig. 3 .

 23 Fig. 2. CZE-UV off-line coupled to MALDI-MS via fractionation for a middle-up characterization of Cetuximab Fc/2 variants. (a) CZE-UV electropherogram. (b) Analysis of CZE-UV fractions by MALDI-MS. CE Experimental conditions: See Fig. 1; MS condition: See Experimental section; sample, IdeS digest of cetuximab (5 µg/µL)

Fig. 4 .

 4 Fig. 4. Fc/2 variants deconvoluted mass spectra of (a) IdeS digest cetuximab, (b) IgGZERO+IdeS digest cetuximab and (c) CP-B+IdeS digest cetuximab. Experimental conditions: 50%/49%/1%, acetonitrile/water/formic acid (v/v/v); sample concentration 1 µM.

Fig. 5 .

 5 Fig. 5. Cetuximab Fc/2 variants deconvoluted mass spectra in (a) denaturing and (b) CZE conditions. Experimental conditions: denaturing conditions, 50%/49%/1%, acetonitrile/water/formic acid (v/v/v); CZE conditions, ammonium acetate 25mM, pH 5.7; sample concentration 1 µM.

Acknowledgments

Authors would like to thank Sciex separations Inc. for lending a CESI8000 system and Dr. M. Anselme from Sciex for his support. The authors would like also to express their gratitude to Dr. E. Wagner-Rousset, MC. Janin-Bussat and O. Colas (Centre d'Immunologie Pierre Fabre, St Julien en Genevois, France) for helpful discussions around antibody structural characterization.

terminal lysine of cetuximab. Sample was treated by carboxypeptidase B (CP-B), removing systematically the C-terminal lysine residues from the two HCs of the antibody [START_REF] Antes | Analysis of lysine clipping of a humanized Lewis-Y specific IgG antibody and its relation to Fc-mediated effector function[END_REF], it was added prior to IdeS digestion step. After CP-B and IdeS treatment, the most intense ion series corresponds to the expected glycoforms of Fc/2-K variants (Fig. 4-c). Moreover, as compared to sample treated only with IdeS (Fig. 4-a), we observe in the deconvoluted mass spectrum the complete loss of Fc/2 glycoforms profile which confirms the total removing of C-terminal lysine on the HC. CZE separation, represented on Fig. 3-C, contains only one peak corresponding to Fc/2 subunits. This peak has been characterized as Fc/2-K variants. Moreover, we observe the absence of the two first peaks compared to the original separation. Despite the fact that the loss of the first is totally justified by the unique presence of Fc/2 variant trimmed from both HC C-terminal lysine, the loss of the second peak proves that the presence of C-terminal lysine residue plays an active role in the separation process of Fc/2 variants. These results also allow to reject hypothetic effects on effective mobilities due to several cumulative PTMs leading to modification of the mAb without MS resolved mass shift. The second peak of the separation of sample without CP-B treatment (Fig. 3A) has been characterized by a nearly equal abundance mixture of Fc/2 and Fc/2-K variants. Absence of this peak in cetuximab separation after CP-B+IdeS digestion (Fig. 3-C) and the fact that no Fc/2-K variants have been detected at that migration time involves a strong relation between Fc/2 and Fc/2-K variants.

Role of Fc/2 aggregates on CZE separation

In this work, we described a protocol based on nanoESI-MS infusion of the different fractions using CESI-MS interface as nanoESI platform. After CZE separation, each collected fraction was systematically dried by air and reconstituted in 2µL of ACN/H20/FA (50/49/1, v/v) to enhance ionization. However, infusion conditions differed from CZE conditions at pH 5.7 and are denaturing by the presence of acetonitrile. Indeed, in these experiments, noncovalent interactions between Fc/2 variants can influence CZE separation at pH 5.7 but protein-protein interaction cannot actually be observed due to the denaturing nanoESI-MS conditions used. To evaluate these phenomena, nanoESI-MS infusion of cetuximab IdeS digest have been prepared using denaturing conditions and using CZE conditions as well. For denaturing conditions, sample has been diluted in 2µL as stated previously while for CZE conditions, sample has been diluted in ammonium acetate 25mM, pH 5.7. Each sample has been reconstituted to a concentration of 1µM. Each sample was infused into the MS using a flow rate of 10 nL/min. Deconvoluted mass spectra in denaturing conditions generated multiple masses corresponding to the expected presence of neutral glycosylation variants of Fc/2 fragments (Fig. 5-a and Table S-1). Mass peak 25359.2 Da corresponding to G0F for Fc/2 and mass peak 25231.1 Da corresponding to G0F for Fc/2-K are in agreement with theoretical masses of these glycoforms (Table S-


Pierre Gilles Lemarié-Rieusset
email: pierregilles.lemarierieusset@univ-evry.fr

Interpolation, extrapolation, Morrey spaces and local energy control for the Navier-Stokes equations

Keywords: Navier-Stokes equations, Morrey spaces, Besov spaces, uniformly locally square integrable functions, weak-strong uniqueness AMS classification : 35K55, 35Q30, 76D05

Interpolation, extrapolation, Morrey spaces and local

energy control for the Navier-Stokes equations.

Under reasonable assumptions, the problem is equivalent to the following integro-differential problem : u = e t∆ u 0 -B( u, u) (t, x) where B( u, v) = t 0 e (t-s)∆ P div ( u ⊗ v) ds

(1)

and P is the Leray projection operator. (See [LR 2, LR 6] for details).

Weak Leray solutions for the Navier-Stokes equations

When u 0 ∈ L 2 , Leray proved existence of solutions u on (0, +∞) × R 3 such that :

• u ∈ L ∞ t L 2 x ∩ L 2 t Ḣ1 x
• lim t→0 + u(t, .) -u 0 2 = 0

• we have the Leray energy inequality

u(t, .) 2 2 + 2 t 0 ∇ ⊗ u 2 2 ds ≤ u 0 2 2
(2) Such solutions are weak solutions : the derivatives in the Navier-Stokes solutions are taken in the sense of distributions. Those solutions (that satisfy the energy inequality (2)) are called Leray weak solutions.

When u 0 ∈ L 2 , Leray's proof of existence of solutions [Le] is based on mollification, energy estimates and compactness arguments :

• he solves ∂ t u ǫ + (ϕ ǫ * u ǫ ). ∇ u ǫ = ∆ u ǫ -∇p ǫ
with div u ǫ = 0 and u ǫ (0, .) = u 0 . Here, ϕ ∈ D, ϕ dx = 1 and ϕ ǫ (x) = 1 ǫ 3 ϕ( x ǫ ). • the solution holds on an interval (0, T ǫ ) where T ǫ depends on ǫ and on u 0 2 and we have the equality

u ǫ (t, .) 2 2 + 2 t 0 ∇ ⊗ u ǫ 2 2 ds = u 0 2 2
• the solution is then global; moreover by Rellich theorem, we find a subsequence that converges strongly in (L 2 t L 2 x ) loc to a Leray solution u

Such solutions (i. e. obtained by this mollification/extraction process) will be called in the following restricted Leray weak solutions.

Restricted Leray solutions satisfy the Leray energy inequality which takes into account the energy on the whole space. But they enjoy as well a pointwise inequality property : for a non-negative locally finite measure µ we have

∂ t (| u| 2 ) + 2| ∇ ⊗ u| 2 = ∆(| u| 2 ) -div ((2p + | u| 2 ) u) -µ (3)
Leray solutions that enjoy the pointwise energy inequality are called suitable Leray solutions [CKN].

Local weak Leray solutions

The pointwise energy inequality allows one [LR 1, LR 2] to develop a theory of weak solutions with infinite energy. Consider u 0 a divergence-free vector field that is uniformly locally square integrable : sup

x 0 ∈R 3 |x-x 0 |<1 | u 0 (x)| 2 dx < +∞.
A local Leray solution on (0, T ) × R 3 is a solution such that

• u ∈ L ∞ t (L 2 uloc ) ∩ (L 2 t Ḣ1
x ) uloc • for all compact subset K of R 3 , lim t→0 + K | u(t, .) -u 0 | 2 dx = 0

• we have the pointwise energy inequality (3).

Local in time existence of restricted local Leray solutions has been proved for a positive T that depends only on u 0 L 2 uloc (see section 8).

The Prodi-Serrin criterion for weak-strong uniqueness

Based on a compactness criterion, the proof of existence of Leray solutions does not provide any clue on the would-be uniqueness of the solution to the Cauchy initial value problem.

A classical case of uniqueness of Leray weak solutions is Serrin's criterion for weak-strong uniqueness [Pr, Se]. If u 0 ∈ L 2 and if the Navier-Stokes equations has a solution u on (0, T ) such that

u ∈ X T = L p t L q x with 2 p + 3 q = 1 and 2 ≤ p < +∞ then if v is a Leray solution we have u = v on (0, T ).
The proof of the criterion is based on the fact that if v is a Leray solution and if u is the mild solution with u X T < +∞, then the difference w = u-v satisfies a Gronwall estimate :

w(t, .) 2 2 + 2 t 0 ∇ ⊗ w 2 2 ds ≤ 2 T 0 | u.( w. ∇ w) dx| ds.
We have (for

2 p + 3 q = 1) u ⊗ w 2 ≤ C u q w 2 p 2 ∇ ⊗ w 3 q 2 so that | u.( w. ∇ w) dx| ≤ u q w 2 p 2 ∇ ⊗ w 1+ 3 q 2 = u q ( w 2 ) 1 p ( ∇ ⊗ w 2 ) 1-1 p .
Thus, we find

2 T 0 | u.( w. ∇ w) dx| ds ≤ C t 0 u p q w 2 2 ds + t 0 ∇ ⊗ w 2 2 ds.
The facts that w(0, .) 2 2 = 0 and w(t, .) 2 2 ≤ C t 0 u p q w 2 2 ds then gives w = 0 on (0, T ).

We may comment a little further in the case 2 < p < +∞. In that case, the bilinear operator B (given by (1)) is bounded on X T = L p t L q x . Thus, we find that the existence of T > 0 and of a solution in L p L q with 2 p + 3 q is equivalent to the existence of T ′ such that e t∆ u 0 ∈ L p L q on (0, T ′ ) (and on (0, +∞), since u 0 ∈ L 2 ). Using the thermic characterization of Besov spaces, we can see that this is equivalent with

u 0 ∈ Ḃ-1+ 3 q q,p
.

Thus, the initial value is not only in L 2 , but it must belong as well to a Besov space with a better regularity than provided by the embedding

L 2 ⊂ Ḃ-3 2 + 3 q q,2 ⊂ Ḃ-3 2 + 3 q q,p
.

3 The Koch and Tataru theorem and T. Barker's question

We may now wonder how to generalize the Prodi-Serrin criterion. It means : given u 0 ∈ L 2 and weak Leray solutions associated to u 0 , find a space X (as large as possible) such that if moreover u 0 ∈ X then we have a solution u ∈ X T for some space X T of functions on (0, T ) × R 3 and such that the existence of a solution in X T implies that any other weak Leray solution is equal to this solution u for 0 < t < T .

The space BMO -1

First of all, we precise which kind of space X we are going to study. The idea is to look at an initial value which generates a solution in some uniqueness class (where uniqueness holds for small solutions). The setting where to construct such solutions is the setting of mild solutions, as introduced by Kato [Ka] : mild solutions are constructed by the Banach contraction principle.

Due to the symmetries of the equations (if u is a solution for initial value u 0 , then λ u(λ 2 t, λ(x -x 0 )) is a solution for the initial value λ u 0 (λ(x -x 0 ))), we look for spaces with norms invariant through the transforms u 0 (.) → λ u(λ(. -x 0 )) (for λ > 0). Moreover, in order to be able to define

B( u, v) = t 0 e (t-s)∆ P div ( u ⊗ v) ds
at least for u = v = e t∆ u 0 (first step of the Picard iteration to find a fixedpoint to u = e t∆ u 0 -B( u, u)), we ask that [0,1]×B(0,1) |e s∆ u 0 (y)| 2 ds dy < +∞.

Thus, we are lead to introduce the space X of distributions v such that

• sup t>0 √ t e t∆ v ∞ < +∞ • sup 0<t,x 0 ∈R 3 t -3/2 t 0 B(x 0 , √ t) |e s∆ v(y)| 2 dy ds) 1/2
This space X has been identified by Koch and Tataru [KocT] : this is the Triebel-Lizorkin space Ḟ -1 ∞,2 , or equivalently the space BMO -1 = √ -∆ BMO. Moreover, they proved the following theorem :

Theorem 1 For 0 < T ≤ ∞, define u X T = sup 0<t<T √ t u(t, . ∞ + sup 0<t<T,x 0 ∈R 3 (t -3/2 t 0 B(x 0 , √ t) | u(s, y)| 2 dy ds) 1/2
There exists C 0 (which does not depend on T ) such that if T ∈ (0, +∞], if u and v are defined on (0, T ) × R 3 and if

B( u, v) = t 0 e (t-s)∆ P div ( u ⊗ v) ds then B( u, v) X T ≤ C 0 u X T v X T .
Corollary 1 If e t∆ u 0 X T < 1 4C 0 , then the integral Navier-Stokes equations have a solution on (0, T ) such that u X T ≤ 2 e t∆ u 0 X T . This is the unique solution such that u X T ≤ 1 2C 0 .

A special case of initial data that leads to a solution in some X T is given by the subspace VMO -1 of BMO -1 .

Definition 1 VMO -1 is the closure of compactly supported functions in BMO -1 .

If u 0 ∈ VMO -1 , then lim T →0 e t∆ u 0 X T = 0. Remark that we have the embedding Ḃ-1+ 3 q q,p ⊂ VMO -1 for 2 < p < +∞ and 2 p + 3 q = 1. As a matter of fact, we may consider VMO -1 as a limit case for the scale of spaces Ḃ-1+ 3 q q,p . Thus, Barker [Ba] raised the following question :

Question 1 If u 0 belongs to L 2 ∩ VMO -1
, does there exists a positive time T such that every weak Leray solution of the Cauchy problem for the Navier-Stokes equations with u 0 as initial value coincide with the mild solution in X T ?

If u 0 ∈ L 2 ∩VMO -1 and if e t∆ u 0 X T ≤ 1 4C 0 , then if u is a restricted Leray solution of the Navier-Stokes solutions with initial value u 0 , then v X T ≤ 2 e t∆ u 0 X T . In particular, we have uniqueness of restricted Leray solutions on (0, T ).

As a matter of fact, this proof of local uniqueness of restricted weak Leray solutions holds for a slightly more general class :

Definition 2 BMO -1 0 is the space of distributions u 0 in BMO -1 such that lim T →0
e t∆ u 0 X T = 0.

In the following, we will focus on the hypothesis u 0 ∈ L 2 ∩ BMO -1 0 and on the issue of uniqueness for Leray solutions.

The limiting case

Up to now, we don't know how to prove local uniqueness of the Leray solutions when the initial value u 0 belongs to L 2 ∩ BMO -1 0 . What we know for sure is that the mild solution u in X T belongs to L ∞ ((ǫ, T ) × R 3 ) for every positive ǫ ∈ (0, T ). Moreover, u is a weak Leray solution and for every other Leray solution v and for ǫ > 0, we have

∂ t ( u. v) = u.∂ t v + ∂ t u. v which gives for 0 < ǫ < t < T u(t, x). v(t, x) dx = u(ǫ, x). v(ǫ, x) dx -2 t ǫ ∇ ⊗ u. ∇ ⊗ v dx ds - t ǫ u.( v. ∇ v) + v.( u. ∇ u) dx ds From u.( v. ∇ v)+ v.( u. ∇ u) dx = u.( v. ∇( v-u))+( v-u).( u. ∇ u) dx = u ( v-u). ∇( v-u) dx
and letting ǫ go to 0, we get

u(t, x). v(t, x) dx = u 0 2 2 -2 t 0 ∇ ⊗ u. ∇ ⊗ v dx ds -lim ǫ→0 t ǫ u ( v -u). ∇( v -u) dx ds and (letting v = u) u(t, .) 2 2 = u 0 2 2 -2 t 0 ∇ ⊗ u 2 2 ds.
Combining those two equalities with the Leray energy inequality for v

v(t, .) 2 2 + 2 t 0 ∇ ⊗ v 2 2 ds ≤ u 0 2 2
we get the following inequality for w = u -v :

w(t, .) 2 2 + 2 t 0 ∇ ⊗ w 2 2 ds ≤ 2 T 0 | u.( w. ∇ w) dx| ds. (4)
As a matter of fact, the key ingredient in Prodi-Serrin's criterion is the estimation of the integral

I( u, w) = t 0 u(. w. ∇ w) dx ds but, if w = v -u
with v a Leray solution and u the mild solution in X T , we don't even know whether I( u, w) is finite.

In the limiting case of Prodi and Serrin, (for p = 2 and q = +∞), we write

w(t, .) 2 2 +2 t 0 ∇⊗ w 2 2 ds ≤ 2 t 0 t 0 u(. w. ∇ w) dx ds ≤ 2 t 0 u⊗ w 2 ∇⊗ w 2 ds
and get

w(t, .) 2 2 ≤ t 0 u 2 ∞ w 2 2 ds. (5)
Of course, we may conclude under the assumption that u ∈ L 2 t L ∞ x . Actually, we shall not be interested in measurabilty issues for functions with values in a non-separable space such as L ∞ (i.e. in Bochner measurability for instance), as we are dealing with locally integrable functions for the Lebesgue measure dt dx on (0, T )×R 3 . Thus, for almost every t the quantity u(t, .) ∞ will be wel-defined as a measurable function of t, and u ∈ L 2 t L ∞ x will simply mean that

T 0 u(t, .) 2 L ∞ (dx) dt < +∞. (See [LR 2] for details.) If u ∈ L 2 t L ∞
x on (0, T ) × R 3 , the Prodi-Serrin criterion proves that every weak Leray solution v on (0, T ) is equal to the mild solution u. (This has even be extended to the case u ∈ L 2 t BMO x by Kozono and Tanyuchi [KozT]). But it is not easy to translate the condition that u ∈ L 2 t L ∞

x into an equivalent assumption on u 0 . The problem comes from the fact that the bilinear operator B is not bounded on L 2 t L ∞ x . On the other hand, if we only assume u 0 ∈ L 2 ∩ BMO -1 0 , we only know the inequality u(t, .) ∞ ≤ u X T o(1) √ t . We find an integrability issue near t = 0. To check that this is actually an issue, consider the following example : take ω a divergence-free vector field in the Schwartz class such that the Fourier transform of ω is compactly supported in the annulus 1 < |ξ| < 2; define

u 0 (x) = +∞ j=0 2 j 1 √ 1 + j ω(2 j x); we have u 0 ∈ L 2 ; u 0 ∈ Ḃ-1+ 3 q q,p ⊂ Ḃ-1 ∞,p (with 2 < p < +∞ and 2 p + 3 q = 1) but u 0 / ∈ Ḃ-1 ∞,2 ; as the bilinear operator B is bounded on L p t L q x ∩ L 2 t L ∞ x , if we assume that the mild solution u ∈ L p t L q x belongs to L 2 t L ∞ x , we would find that e t∆ u 0 ∈ L 2 t L ∞
x , and thus u 0 ∈ Ḃ-1 ∞,2 ; thus, we have

T 0 u 2 ∞ dt = +∞.

Barker's theorem

In this section, we shall sketch the proof of Barker [Ba], as we shall extend it in Section 7 to the case of Besov-Morrey spaces.The main idea in the recent paper of Barker is the following one : if we want to use only the inequality

u(t, .) ∞ ≤ u X T o(1) √ t
to deal with the Gronwall inequality (5), we need to assume more than w ∈ L ∞ t L 2

x . Indeed, we have the easy following lemma :

Lemma 1 Let δ > 0. Let A and B be locally bounded non-negative measurable functions on (0, T ] such that

lim t→0 tA(t) = 0 and sup 0<t<T t -δ B(t) = 0.
If we have moreover, for all t ∈ (0, T ],

B(t) ≤ t 0 A(s)B(s) ds then B = 0.
Proof : We have

B(t) ≤ t δ δ sup 0<s<t sA(s) sup 0<σ<t σ -δ B(σ)
so that B = 0 on (0, T 0 ] as long as sup 0<s<T 0 sA(s) < δ. For t > T 0 , we then write B(t) ≤ sup T 0 <s<T A(s) t T 0 B(s) ds and we find B = 0. ⋄ As w = v -u = ( v -e t∆ u 0 ) -( u -e t∆ u 0 ), the extra information on w 2 will be provided by the following lemma :

Lemma 2 Let u 0 be a divergence-free vector field with u 0 ∈ L 2 and let v be a weak Leray solution of the Navier-Stokes equations with initial value u 0 . If moreover

u 0 ∈ [L 2 , B -γ ∞,∞ ] θ,∞ for some -1 < -γ < 0 and 0 < θ < 1
then there exists δ > 0 such that

sup t>0 t -δ v(t, .
) -e t∆ u 0 2 < +∞.

Proof : Assume u 0 ∈ [L 2 , B -γ ∞,∞ ] θ,∞ . For 0 < ǫ < 1, split u 0 in α ǫ + β ǫ with α ǫ 2 ≤ C 0 ǫ θ and β ǫ Ḃ-γ ∞,∞ ≤ C 0 ǫ θ-1
where C 0 does not depend on ǫ (but depends on u 0 ).

We have a solution U ǫ for the Navier-Stokes equations with initial value

β ǫ such that U ǫ (t, .) ∞ ≤ C 1 t -γ 2 ǫ θ-1 on an interval (0, T ǫ ) with T 1-γ 2 ǫ ǫ θ-1 = C 2 . Moreover β ǫ = u 0 -α ǫ ∈ L 2 and we find that sup 0<t<Tǫ U ǫ 2 ≤ C 3 and that U ǫ -e t∆ β ǫ 2 ≤ C 4 t (1-γ)/2 ǫ θ-1 . Since U ǫ is in L 2 t L ∞
x on every bounded interval, we get

U ǫ (t, x). v(t, x) dx = β ǫ (x). u 0 (x) dx -2 t 0 ∇ ⊗ U ǫ . ∇ ⊗ v dx ds - t 0 U ǫ ( v -U ǫ ). ∇( v -U ǫ ) dx ds and U ǫ (t, .) 2 2 = β ǫ 2 2 -2 t 0 ∇ ⊗ U ǫ 2 2 ds.
Combining those two equalities with the Leray energy inequality for v v(t, .) 2 2 + 2

t 0 ∇ ⊗ v 2 2 ds ≤ u 0 2 2
we get the following inequality for W ǫ = v -U ǫ :

W ǫ (t, .) 2 2 + 2 t 0 ∇ ⊗ W ǫ 2 2 ds ≤ α ǫ 2 2 + 2 T 0 | U ǫ .( W ǫ . ∇ W ǫ ) dx| ds.
Thus, we get

W ǫ (t, .) 2 2 ≤ C 2 0 ǫ 2θ + C 2 1 ǫ 2(θ-1) t 0 s -γ W ǫ (s, .) 2 2 ds so that W ǫ (t, .) 2 2 ≤ C 2 0 ǫ 2θ e C 2 1 ǫ 2(θ-1) t 1-γ 1-γ .
Now, for τ < 1, take ǫ = τ µ with 1-γ 2 + µ(θ -1) > 0. We find that, for 0 < t < T ǫ with

T ǫ = C 2 1-γ 2 ǫ 2(1-θ) 1-γ = C 2 1-γ 2 τ µ 2(1-θ) 1-γ [ where µ 2(1 -θ) 1 -γ < 1],
we have the inequality

v -e t∆ u 0 2 ≤ α ǫ 2 + W ǫ 2 + U ǫ -e t∆ β ǫ 2 ≤C 0 ǫ θ (1 + e C 2 1 2(1-γ) t 1-γ T 1-γ ǫ ) + C 4 t 1-γ 2 ǫ θ-1 If τ is small enough, we have τ < T ǫ and we find v(τ, .) -e τ ∆ u 0 2 ≤ C 0 τ µθ (1 + e C 2 1 2(1-γ ) + C 4 τ 1-γ 2 +µ(θ-1)
The lemma is proved. ⋄ Barker's theorem then reads as :

Theorem 2 Let u 0 be a divergence-free vector field with u 0 ∈ L 2 and let v be a weak Leray solution of the Navier-Stokes equations with initial value u 0 . If moreover

u 0 ∈ BMO -1 0 ∩ Ḃ-s q,∞ with 3 < q < +∞ and -s > -1 + 2 q
then there exists T > 0 such that if v is a Leray solution and if u is the mild solution with u X T < +∞, then u = v on (0, T ).

Remark We have the embeddings

L 2 ⊂ Ḃ-3 2 + 3 q q,2 ⊂ Ḃ-3 2 + 3 q q,
∞ , so that the information conveyed by the hypothesis u 0 ∈ Ḃ-s q,∞ is interesting only for the high frequencies of u 0 . Moreover the embeddings

L 2 = Ḃ0 2,2 and BMO -1 ⊂ Ḃ-1 ∞,∞ gives that L 2 ∩ BMO -1 0 ⊂ Ḃ-1+ 2 q q,q ⊂ Ḃ-1+ 2 q
q,∞ . Thus, the information conveyed by the hypothesis u 0 ∈ Ḃ-s q,∞ is not contained in the assumption

u 0 ∈ L 2 ∩ BMO -1 0 . Finally, if -s > -1 + 3 q = 2 p , then we have L 2 ∩ BMO -1 0 ∩ Ḃ-s q,∞ ⊂ Ḃ-1+ 2 q q,q ∩ Ḃ-s q,∞ ⊂ Ḃ-1+ 3 q q,p .
Thus, the theorem is interesting only in the range -1 + 2 q < -s ≤ -1 + 3 q , which corresponds to the gap between L 2 ∩ BMO -1 0 . (where loca'luniqueness is conjectured to hold) and Ḃ-1+ 3 q q,p (for which the Prodi-Serrin criterion shows that local uniqueness holds).

A further remark is that we have the embedding Ḃ-1+ 3 q q,∞ ⊂ BMO -1 , so that we have a Prodi-Serrin criterion with u ∈ L p t L q x (with 2 p + 3 q = 1) replaced with sup 0<t<T t 1 p u q < +∞ and lim t→0 t 1 p u(t, .) q = 0.

Proof :

The first step is the use of interpolation inequalities in order to be able to check that u 0 fulfills the assuptions of Lemma 2.

• Since u 0 ∈ L 2 = Ḃ0 2,2 and u 0 ∈ BMO -1 ⊂ Ḃ-1 ∞,∞ , we have u 0 ∈ Ḃ-1+ 2 q q,q
.

• Since u 0 ∈ Ḃ-1+ 2 q q,q ∩ Ḃ-s q,∞ , we have for -1 + 2 q < -σ < -s, u 0 ∈ Ḃ-σ q,1 .

• The Besov space Ḃ-σ q,1 is embedded in the Sobolev space Ẇ -σ,q .

• For q < r < ∞, we have Ẇ -σ,q = [L 2 , Ẇ -δ,r ] [θ] with 1 q = 1-θ 2 + θ r and -s = -θδ (complex inteerpolation)

• Since Ẇ -δ,r ⊂ Ḃ-1+ 3 r ∞,∞ , we have [L 2 , Ẇ -δ,r ] [θ] ⊂ [L 2 , Ḃ-δ-3 r ∞,∞ ] θ,∞ with -δ = - s θ = -s 1 2 -1 r 1 2 -1 q = - (-s) -1 + 2 q + O( 1 r )
so that -δ -3 r > -1 for r large enough We may now end the proof : recall that if v is a Leray solution and if u is the mild solution with u X T < +∞, then the difference w = u -v satisfies a Gronwall estimate :

w(t, .) 2 2 ≤ t 0 u 2 ∞ w 2 2 ds.
By Lemma 2, we have v(t, .) -e t∆ u 0 2 = O(t δ ) and u(t, .) -e t∆ u 0 2 = O(t δ ) for some positive δ. On the other hand, we know that u(t, .) ∞ = o( 1 √ t ). Using Lemma 1, we find that w = 0, and v = u.⋄

The Prodi-Serrin criterion for Besov-Morrey spaces

Morrey spaces provide a natural tool for extending the Prodi-Serrin criterion.

Definition 3 For 1 < r ≤ q < +∞, we define the Morrey space Ṁr,q as the space of Lebesgue measurable functions

f on R 3 such that sup R>0,x 0 ∈R 3 R 3 q -3 r ( B(x 0 ,R) |f (x)| r dx) 1/r = f Ṁ r,q < +∞.
Similarly, the space Ṁ1,q is the space of locally finite Borelian (signed) measure µ such that

sup R>0,x 0 ∈R 3 R 3 q -3 ( B(x 0 ,R) d|µ|) = µ Ṁ 1,q < +∞.
Remark : For absolutely continuous measures dµ = f dx with f ∈ L 1 loc , we have

µ Ṁ 1,q = sup R>0,x 0 ∈R 3 R 3 q -3 ( B(x 0 ,R) |f (x)| dx).
The key inequality in the proof of the Prodi-Serrin criterion was the inequality (for all w ∈ H 1 )

uw 2 ≤ C u q w 2 p 2 ∇w 3 q 2 with 2
p + 3 q = 1 and 3 < q < +∞. If we want to replace this inequality by a more general inequality

uw 2 ≤ N(u) w 2 p 2 ∇w 3 q

2

(again with 2 p + 3 q = 1 and 3 < q < +∞), then we proved in [LR 3] that the existence of a finite N(u) is equivalent to the fact that u ∈ Ṁ2,q , and moreover that N(u) ≈ u Ṁ 2,q .

This leads to the following easy extension of the Prodi-Serrin criterion :

Theorem 3 If u 0 ∈ L 2 and if the Navier-Stokes equations has a solution u such that u ∈ L p t Ṁ2,q

x with 2 p + 3 q = 1 and 3 < q < +∞ then if v is a Leray solution we have u = v on (0, T ).

If 3 < q < +∞, the existence of T > 0 and of a solution in L p Ṁ2,q with 2 p + 3 q is equivalent to the existence of T ′ such that e t∆ u 0 ∈ L p Ṁ2;q on (0, T ′ ) (and on (0, +∞), since u 0 ∈ L 2 ), thus with

u 0 ∈ Ḃ-1+ 3 q Ṁ 2,q ,p .
This Besov-Morrey space has been introduced in 1994. by Kozono and Yamazaki [KoY]. It is easy to check that, for 2 < p < +∞ and 2 p + 3 q = 1, we have the inequality e t∆ u 0 X T ≤ C q ( T 0 e t∆ u 0 p Ṁ 2,q dt) 1:p so that we have the embedding Ḃ-1+ 3 q Ṁ 2,q ,p ⊂ BMO -1 0 for 2 < p < +∞ and

2 p + 3 q = 1.

Barker's theorem and Besov-Morrey spaces

We shall extend Barker's theorem.

Theorem 4 If • u 0 ∈ L 2 ∩ BMO -1 0 • 3 < q < +∞, -s > -1 + 2 q and u 0 ∈ Ḃ-s Ṁ 1,q ,∞
then there exists T > 0 such that if v is a suitable Leray solution and if u is the mild solution with u X T < +∞, then u = v on (0, T ).

Proof : As we shall see, the proof is very similar to Barker's proof for Theorem 2 [Ba]. However, we shall meet some technical issues. We sketch the proof :

• u 0 ∈ L 2 = Ḃ0 2,2 and u 0 ∈ BMO -1 ⊂ Ḃ-1 ∞,∞ , thus u 0 ∈ Ḃ-1+ 2 q q,q • u 0 ∈ Ḃ-1+ 2 q q,q ∩ Ḃ-s Ṁ 1,q ,∞ thus u 0 ∈ Ḃ-σ Ṁ p,q ,∞ for 1 < p < q, 1 p = 1 -θ + θ q and -σ = -s(1 -θ) + θ(-1 + 2 q ) > -1 + 2 q .
We shall take p > 2.

• as p < q, we have Ḃ-1+ 2 q q,q ⊂ Ḃ1+ 2 q q,p . Thus, for -1

+ 2 q < -γ < -σ, Ḃ-1+ 2 q q,q ∩ Ḃ-σ Ṁ p,q ,∞ ⊂ Ḃ-γ Ṁ p,q ,1 ⊂ Ẇ -γ, Ṁ p,q .
We now encounter our first problem. We can no longer write W -γ, Ṁ p,q as a subspace of an interpolate space between L 2 and Ḃ-1+δ ∞,∞ . More precisely, let us assume Ḃ-γ

Ṁ p,q ,1 ⊂ [L 2 , Ḃ-1+δ ∞,∞ ] θ,∞
; by homogeneity of the norms, we must have -γ -

3 q = -(1 -θ) 3 2 -θ(1 -δ). We have [L 2 , Ḃ-1+δ ∞,∞ ] θ,∞ ⊂ Ḃ-θ(1-δ) L r,∞ ,∞ with r = 2 1-θ . In particular, for u ∈ [L 2 , Ḃ-1+δ ∞,∞ ] θ,∞
, we have that e ∆ e t∂ 2 3 u goes to 0 in S ′ when t goes to +∞. But if 3p ≤ 2q, if u depends only on (x 1 , x 2 ) and not on x 3 , and if u ∈ Ḃ-γ

Ṁ p, 2q 3 ,1 (R 2 ), then u ∈ Ḃ-γ Ṁ p,q ,1 (R 3
) and e ∆ e t∂ 2 3 u = e ∆ u. Thus, we have a contradiction. We better use complex interpolation and write that

W -γ, Ṁ p,q = [ Ṁ2, 2 p q , Ẇ -ρ, Ṁ r, r p q ] [θ]
for r > p, 1-θ 2 + θ r = 1 p , γ = θρ. (For interpolation of Morrey spaces, see [LR 4, LR 5])

Then, we remark that Ṁ2, 2 p q ⊂ L 2 uloc and write that

[ Ṁ2, 2 p q , Ẇ -ρ, Ṁ r, r p q ] [θ] ⊂ [ Ṁ2, 2 p q , Ẇ -ρ, Ṁ r, r p q ] θ,∞ ⊂ [L 2 uloc , Ḃ-ρ-3p rq ∞,∞ ] θ,∞
In order to finish the proof, we thus need to use the machinery of energy control for suitable local Leray solutions [LR 2,LR 6]. This will be done in the following sections, and we shall finish the proof in Section 10 ⋄

Weak local Leray solutions

We recall basic results for local weak Leray solutions. We endow L 2 uloc with the norm u L 2 uloc = sup

k∈Z 3 uϕ 0 (x -k) 2 ,
where ϕ 0 is a non-negative function in D, suppported in a ball B(0, R 0 ) and such that k∈Z 3 ϕ 0 (x -k) = 1. When u 0 ∈ L 2 uloc , proof of existence of solutions for the Navier-Stokes equations is based on mollification, energy estimates and compactness arguments (for details, see [LR 6], section 14.1) :

• we solve

∂ t u ǫ + (ϕ ǫ * u ǫ ). ∇ u ǫ = ∆ u ǫ -∇p ǫ
with div u ǫ = 0 and u ǫ (0, .) = u 0 . Here, ϕ ∈ D, ϕ dx = 1 and ϕ ǫ (x) = 1 ǫ 3 ϕ( x ǫ ). Here ∇p ǫ is given by the Leray projection :

∇p ǫ = -(ϕ ǫ * u ǫ ). ∇u ǫ + Pdiv ((ϕ ǫ * u ǫ ) ⊗ u ǫ ) .
• the solution holds at least on an interval (0, T ǫ ) where T ǫ depends on ǫ and on u 0 L 2 uloc (T ǫ = min(1, C 0

ǫ 3/2 u 0 2 L 2 uloc
). Moreover, we have the inequalities, for k ∈ Z 3 ,

ϕ 0 (x -k)| u ǫ (t, x)| 2 dx + 2 t 0 ϕ 0 (x -k)| ∇ ⊗ u ǫ (s, x)| 2 dx ds ≤ ϕ 0 (x -k)| u 0 (t, x)| 2 dx + C 1 t 0 |x-k|≤R 0 | u ǫ (s, x)| 2 dx ds + C 2 t 0 |x-k|>5R 0 1 |x -k| 4 | u ǫ (s, x)| 3 dx ds + C 2 t 0 |x-k|<5R 0 | u ǫ (s, x)| 3 dx ds • defining α ǫ (t) = u ǫ (t, .) L 2 uloc and β ǫ (t) = sup k∈Z 3 t 0 ϕ 0 (x -k)| ∇ ⊗ u ǫ (s, x)| 2 dx ds 1/2
, we get the inequality

ϕ 0 (x -k)| u ǫ (t, x)| 2 dx + t 0 ϕ 0 (x -k)| ∇ ⊗ u ǫ (s, x)| 2 dx ds ≤ α ǫ (0) 2 + 1 2 β ǫ (t) 2 + C 3 t 0 α ǫ (s) 2 ds + C 3 t 0 α ǫ (s) 6 ds so that β ǫ (t) 2 ≤ 2α ǫ (0) 2 + 2C 3 t 0 α ǫ (s) 2 ds + 2C 3 t 0 α ǫ (s) 6 ds
and finally

α ǫ (t) 2 ≤ 2α ǫ (0) 2 + 2C 3 t 0 α ǫ (s) 2 ds + 2C 3 t 0 α ǫ (s) 6 ds
Thus, as long as 8C 3 t < 1 and 128 C 3 t u 0 4 L 2 uloc < 1, we find that α ǫ (t) ≤ 2 u 0 L 2 uloc and β ǫ (t) ≤ 2 u 0 L 2 uloc . • the solution is then defined on (0, min( 1 8C 3 ,

1 128 C 3 u 0 4 L 2 loc
) and controlled independently from ǫ. By Rellich theorem, wre find a subsequence that converges strongly in (L 2 t L 2 x ) loc to a suitable local Leray solution u An important point is the following one : assume moreover that u 0 ∈ BMO -1 0 and that e t∆ u 0 X T < 1 4C 0 (where C 0 is the constant of Theorem 1) then u ǫ is defined at least on (0, T ) and u ǫ X T ≤ 2 e t∆ u 0 X T . As T does not depend on ǫ, we see that the local Leray solution u satisfies u ∈ X S with S = min(T, 1 8C 3 ,

1 128 C 3 u 0 4 L 2 loc
) and u X S ≤ 2 e t∆ u 0 X T .

Similarly, if we assume that u 0 ∈ Ḃ-γ ∞,∞ with -1 < -γ < 0, then u ǫ is defined at least on (0, T ) where

T = C u 0 2 1-γ Ḃ-γ ∞,∞
and sup 0<t<T t γ 2 u(t, .) ∞ ≤ 2 sup 0<t<T t γ 2 e t∆ u 0 ∞ . As T does not depend on ǫ, we see that the local Leray solution u satisfies the inequality sup 0<t<S t γ 2 u(t, .) ∞ < +∞ where S = min(T, 1 8C 3 ,

1 128 C 3 u 0 4 L 2 loc
).

Comparison of local weak Leray solutions

If u and v are two local weak Leray solutions, on (0, T ) with initial values u 0 and v 0 , we would like to be able to estimate u(t, .)-v(t, .) from the estimation of u 0 -v 0 . This can be done only when at least one of the solutions is regular enough. We shall assume that u ∈ L 2 t L ∞ x . We sketch the computations described in [LR 6], section 14.4.

Define

w = u -v, α(t) = w(t, .) L 2 uloc and β(t) = sup k∈Z 3 t 0 ϕ 0 (x -k)| ∇ ⊗ w(s, x)| 2 dx ds 1/2
, Using the suitability of v and the regularity of v, we find (for 0 < t < min(1, T ))

ϕ 0 (x -k)| w(t, x)| 2 dx + t 0 ϕ 0 (x -k)| ∇ ⊗ w(s, x)| 2 dx ds ≤ α(0) 2 + 1 2 β(t) + C 1 t 0 α(s) 2 ds + C 2 t 0 α(s) 6 ds + C 3 t 0 u(s, .) 2 ∞ α(s) 2 ds
where the constants C i do not depend on T , u, nor on v. Finally, we find

α(t) 2 ≤2α(0) 2 + 2(C 1 + C 3 ( u L ∞ t L 2 uloc + v L ∞ t L 2 uloc ) 4 ) t 0 α(s) 2 ds + 2C 3 t 0 u(s, .) 2 ∞ α(s) 2 ds. (6)
We have the same estimate even if u is not intregrable near t = 0. Let us only assume that u ∈ L 2 t L ∞ x on every (ǫ, T ) with ǫ > 0. Considering a time t 0 > 0 which a Lebesgue point for the functions t → ϕ(x -k)| u(t, x)| 2 dx and t → ϕ(x -k)| u(t, x)| 2 dx, we have that u and v are local weak Leray solutions on (t 0 , T ) with initial values u(t 0 , .) and v(t 0 , .). Thus, we shall find that, for t > t 0

ϕ 0 (x -k)| w(t, x)| 2 dx ≤2 ϕ 0 (x -k)| w(t 0 , x)| 2 dx + 2(C 1 + C 3 ( u L ∞ t L 2 uloc + v L ∞ t L 2 uloc ) 4 ) t t 0 α(s) 2 ds + 2C 3 t t 0 u(s, .) 2 ∞ α(s) 2 ds. (7)
It is then enough to let t 0 go to 0 and then take the supremum with respect to k.

Proof of Theorem 4

We may now finish the proof. We consider two solutions of the Navier-Stokes equations with initial value

u 0 ∈ L 2 ∩ BMO -1 0 ∩ Ḃ-s Ṁ 1,q ,∞ with -s > -1 + 2 q
(and 3 < q < +∞) : we assume that v is a suitable Leray solution and u is the mild solution in X T . As v is suitable, v is a local Leray solution as well and we may estimate the L 2 uloc of u -w : defining

B(t) = u(t, .) -v(t, .) 2 L 2 uloc and A(t) = 2(C 1 + C 3 ( u L ∞ t L 2 uloc + v L ∞ t L 2 uloc ) 4 ) + 2C 3 u(t, .) 2 ∞ ,
we get

B(t) ≤ t 0 A(s)B(s) ds.
As lim t→0 tA(t) = 0, we shall try to apply Lemma 1. Thus, we shall use interpolation estimates to search for a control of B(t) as O(t -δ ), in the spirit of Lemma 2:. Recall that we have introduced the following numbers :

• p such that 2 < p < 2q 3

• 1 -θ the barycentric coordinate of 1 p in [ 1 q , 1] : 1 p = (1 -θ) + θ 1 q

• -σ the corresponding point in [-1+ 2 q , -s] : -σ = (1-θ)(-s)+θ(-1+ 2 q )

• -γ such that -1 + 2 q < -γ < -σ • r such that p < r < +∞ • 1 -η the barycentric coordinate of 1 p in the segment [ 1 r , 1 2 ] : 1 p = 1-η 2 + η r

• -ρ the corresponding point in [-γ, 0] : -ρ = η(-γ)

We have the following embeddings :

• B 0 2,2 ∩ BMO -1 ⊂ Ḃ-1+ 2 q q,q • Ḃ-1+ 2 q q,q ∩ Ḃ-s Ṁ 1,q ,∞ ⊂ Ḃ-σ Ṁ p,q ,∞ • Ḃ-1+ 2 q q,q
∩ Ḃ-σ Ṁ p,q ,∞ ⊂ Ḃ-γ Ṁ p,q ,1 ⊂ Ẇ -γ, Ṁ p,q .

• W -γ, Ṁ p,q = [ Ṁ2, 2 p q , Ẇ -ρ, Ṁ r, r p q ] [η]

• [ Ṁ2, 2 p q , Ẇ -ρ, Ṁ r, r p q ] [η] ⊂ [ Ṁ2, 2 p q , Ẇ -ρ, Ṁ r, r p q ] η,∞ ⊂ [L 2 uloc , Ḃ-ρ-3p Thus far, we have seen that u 0 ∈ [L 2 uloc , Ḃ-λ ∞,∞ ] η,∞ for some η ∈ (0, 1) and some λ ∈ (0, 1). We shall now estimate v -e t∆ u 0 when v is a weak local Leray solution. For 0 < ǫ < 1, split u 0 in α ǫ + β ǫ with α ǫ L 2 uloc ≤ C 0 ǫ η and β ǫ Ḃ-λ ∞,∞ ≤ C 0 ǫ η-1 where C 0 does not depend on ǫ (but depends on u 0 ).

As β ǫ = u 0 -α ǫ , we have β ǫ L 2 uloc ≤ u 0 L 2 uloc + C 0 (an estimation which does not depend on ǫ) and we know that we have a (restricted) weak Leray solution of the Navier-Stokes equations U ǫ with initial value β ǫ such that sup 0<t<T 0 U ǫ (t, .) L 2 uloc ≤ C 1 , where T 0 and C 1 depends only on u 0 L 2 uloc (and not on ǫ).

As β ǫ ∈ Ḃ-λ ∞,∞ ,, we have as well that U ǫ (t, .) ∞ ≤ C 2 t -λ 2 ǫ η-1 on an interval (0, T ǫ ) with T 1-λ 2 ǫ ǫ η-1 = C 3 It is then easy to check that U ǫe t∆ β ǫ L 2 loc ≤ C 4 t (1-λ)/2 ǫ η-1 . Using our results on comparison of suitable local Leray solutions, we find that we get the following inequality for W ǫ = v -U ǫ and A ǫ (t) = sup k∈Z 3 ϕ 0 (x -k)| W ǫ (t, .x)| 2 dx :

A ǫ (t) 2 ≤ 2 α ǫ 2 L 2 uloc + C 5 t 0 A ǫ (s) 2 ds + C 6 t 0 u(s, .) 2 ∞ A ǫ (s) 2 ds. (8)
Thus, we get A ǫ (t) ≤ C 7 ǫ 2η + C 8 t 0 A ǫ (s) 2 ds + C 9 ǫ 2(η-1) t 0 s -λ A ǫ (s) 2 ds.

so that A ǫ (t) ≤ C 7 ǫ 2η e C 8 t e C 9 ǫ 2(η-1) t 1-λ 1-λ .

  rq ∞,∞ ] η,∞If we take r very large, we have η = 1 -2 p + o(1) and

Now, for τ < 1, take ǫ = τ µ with 1-λ 2 + µ(η -1) > 0. We find that, for 0 < t < min(T ǫ , T 0 ) with

we have the inequality

If τ is small enough, we have τ < T ǫ and we find

The theorem is proved.