final LocalDate today = LocalDate.of(2015, 11, 20);
final LocalDate nextSunday = today.with(next(SUNDAY));
final LocalDate thisPastSunday = today.with(previous(SUNDAY));

import static java.time.DayOfWeek.SUNDAY;
import static java.time.temporal.TemporalAdjusters.next;
import static java.time.temporal.TemporalAdjusters.previous;

Calendar c=Calendar.getInstance();
c.set(Calendar.DAY_OF_WEEK,Calendar.SUNDAY);
c.set(Calendar.HOUR_OF_DAY,0);
c.set(Calendar.MINUTE,0);
c.set(Calendar.SECOND,0);
DateFormat df=new SimpleDateFormat("EEE yyyy/MM/dd HH:mm:ss");
System.out.println(df.format(c.getTime()));      // This past Sunday [ May include today ]
c.add(Calendar.DATE,7);
System.out.println(df.format(c.getTime()));      // Next Sunday
c.add(Calendar.DATE,7);
System.out.println(df.format(c.getTime()));      // Sunday after next

Sun 2010/12/26 00:00:00
Sun 2011/01/02 00:00:00
Sun 2011/01/09 00:00:00

DateFormat df = DateFormat.getDateTimeInstance(DateFormat.FULL, DateFormat.FULL);
Calendar now = new GregorianCalendar();
Calendar start = new GregorianCalendar(now.get(Calendar.YEAR),
        now.get(Calendar.MONTH), now.get(Calendar.DAY_OF_MONTH) );

while (start.get(Calendar.DAY_OF_WEEK) != Calendar.SUNDAY) {
    start.add(Calendar.DAY_OF_WEEK, -1);
}

Calendar end = (Calendar) start.clone();
end.add(Calendar.DAY_OF_MONTH, 7);

System.out.println(df.format(now.getTime()) );
System.out.println(df.format(start.getTime()) );
System.out.println(df.format(end.getTime()) );

// Get the date today, and then select midnight of the first day of the week
// Joda uses ISO weeks, so all weeks start on Monday.
// If you want to change the time zone, pass a DateTimeZone to the method toDateTimeAtStartOfDay()
DateTime midnightToday = new LocalDate().toDateTimeAtStartOfDay();
DateTime midnightMonday = midnightToday.withDayOfWeek( DateTimeConstants.MONDAY );

// If your week starts on Sunday, you need to subtract one.  Adjust accordingly.
DateTime midnightSunday = midnightMonday.plusDays( -1 );

DateTime midnightNextSunday = midnightSunday.plusDays( 7 );
DateTime midnightSundayAfter = midnightNextSunday.plusDays( 7 );

Interval thisWeek = new Interval( midnightSunday, midnightNextSunday );
Interval nextWeek = new Interval( midnightNextSunday, midnightSundayAfter );

if ( thisWeek.contains( someDate.getTime() )) System.out.println("This week");
if ( nextWeek.contains( someDate.getTime() )) System.out.println("Next week");

Date today = new Date(2014, 7, 1);      // assume today is 2014-07-01
Date previousSunday = today.previousOrSame(DayOfWeek.SUNDAY);   // 2014-06-29
Date nextSunday = today.next(DayOfWeek.SUNDAY);                 // 2014-07-06

@Test
public void testThisAndNextWeek() throws Exception {
    GregorianCalendar lastWeekCal = new GregorianCalendar(2010,
            Calendar.DECEMBER, 26);
    int lastWeek = lastWeekCal.get(Calendar.WEEK_OF_YEAR);

    GregorianCalendar nextWeekCal = new GregorianCalendar(2011,
            Calendar.JANUARY, 4);
    int nextWeek = nextWeekCal.get(Calendar.WEEK_OF_YEAR);

    GregorianCalendar todayCal = new GregorianCalendar(2010,
            Calendar.DECEMBER, 27);
    int currentWeek = todayCal.get(Calendar.WEEK_OF_YEAR);

    assertTrue(lastWeekCal.before(todayCal));
    assertTrue(nextWeekCal.after(todayCal));
    assertEquals(51, lastWeek);
    assertEquals(52, currentWeek);
    // New Year.. so it's 1
    assertEquals(1, nextWeek);
}

LocalDate date = ...;
    LocalDate newWeekDate = date.plusDays(1);
    while (newWeekDate.getDayOfWeek() != DayOfWeek.SATURDAY &&
           newWeekDate.getDayOfWeek() != DayOfWeek.SUNDAY) {
        newWeekDate = date.plusDays(1);
    }