### Description

Going to Airflow web UI, Browse>DAG Run displays the list of runs, but there is no way to search or filter based on the text in the "Note" column.

### Use case/motivation

It is possible to do a free text search for the "Run Id" field. The Note field may contain pieces of information that may be relevant to find, or to filter on the basis of these notes.

### Related issues

Sorting by Note in List Dag Run fails: 
https://github.com/apache/airflow/issues/30041

### Are you willing to submit a PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

if request.args.get("reset_tags") is not None:
            flask_session[FILTER_TAGS_COOKIE] = None
            
            return redirect(url_for("Airflow.index"))
        cookie_val = flask_session.get(FILTER_TAGS_COOKIE)
        if arg_tags_filter:
            flask_session[FILTER_TAGS_COOKIE] = ",".join(arg_tags_filter)
        elif cookie_val:
            
            return redirect(url_for("Airflow.index", tags=cookie_val.split(",")))
        if arg_status_filter is None:
            cookie_val = flask_session.get(FILTER_STATUS_COOKIE)
            if cookie_val:
                arg_status_filter = cookie_val
            else:
                arg_status_filter = "active" if hide_paused_dags_by_default else "all"
                flask_session[FILTER_STATUS_COOKIE] = arg_status_filter
        else:
            status = arg_status_filter.strip().lower()
            flask_session[FILTER_STATUS_COOKIE] = status
            arg_status_filter = status
        dags_per_page = PAGE_SIZE
        start = current_page * dags_per_page
        end = start + dags_per_page
        
        filter_dag_ids = get_airflow_app().appbuilder.sm.get_accessible_dag_ids(g.user)
        with create_session() as session:
            
            dags_query = session.query(DagModel).filter(~DagModel.is_subdag, DagModel.is_active)
            if arg_search_query: