at first sight you did not provide an output file to write the log info the option -trimlog takes a file name as argument (now it will indeed overwrite one of your input files)

you also need to provide the output as <paired out 1> <unpaired out 1> <paired out 2> <unpaired out 2>

And I understand you want to speed up the analysis running it multi-core but 140 is seriously over the top, it will normally run at ok speed using 4-6-10 cores

You have to pass the full path to both R1 and R2 files, not just for the R1 file as you are doing. Also, I don't know if the rest of your command is correct, as I don't use Trimmomatic, but I believe it isn't, as the manual states it needs either a --baseout parameter, or 4 file names for output files:

Paired-end mode requires 2 input files (for forward and reverse reads) and 4 output files (for forward paired, forward unpaired, reverse paired and reverse unpaired reads).

So the order of your output files won't be what you expect. Here is what I think is the correct command:

java -jar ~/projects/algae/ABY2/Trimmomatic/Trimmomatic-0.38/trimmomatic-0.38.jar PE 
  -threads 140 -phred33 -trimlog KMM2_log.txt \
  ~/projects/algae/ABY2/sequencing/KMM2_analysis/KMM2-A_R1_001.fastq \
  ~/projects/algae/ABY2/sequencing/KMM2_analysis/KMM2-A_R2_001.fastq \
  KMM2_output_forward_paired_trimmed.fastq KMM2_output_forward_unpaired_trimmed.fastq \
  KMM2_output_reverse_paired_trimmed.fastq KMM2_output_reverse_unpaired_trimmed.fastq \
  ILLUMINACLIP:TruSeq3-PE.fa:2:30:10 LEADING:3 TRAILING:3 SLIDINGWINDOW:4:15 MINLEN:36

edit: lieven.sterck is right, you also forgot to provide a name to the log file. I edited my suggestion to fix this as well.